We consider the following Problem:
Strict Trs:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)
, f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)
, g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)
, g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)
, h(x, x) -> x}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)
, f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)
, g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)
, g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)
, h(x, x) -> x}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4, x5) = [1 1] x1 + [1 1] x2 + [0 0] x3 + [1 1] x4 + [0 0] x5 + [1]
[0 0] [0 0] [1 1] [0 0] [1 1] [1]
a() = [0]
[0]
g1(x1, x2, x3, x4) = [1 0] x1 + [0 1] x2 + [1 0] x3 + [0 0] x4 + [1]
[0 1] [0 0] [0 1] [0 0] [1]
g2(x1, x2, x3, x4) = [1 1] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [1]
[0 0] [1 1] [0 0] [0 0] [1]
h(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 0] [1 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)
, f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)
, g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)
, h(x, x) -> x}
Weak Trs:
{ g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {h(x, x) -> x}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4, x5) = [1 1] x1 + [1 1] x2 + [0 0] x3 + [1 1] x4 + [0 0] x5 + [1]
[0 0] [0 0] [1 1] [0 0] [1 1] [1]
a() = [0]
[0]
g1(x1, x2, x3, x4) = [1 0] x1 + [0 1] x2 + [1 0] x3 + [0 0] x4 + [1]
[0 1] [1 0] [0 1] [0 0] [1]
g2(x1, x2, x3, x4) = [1 1] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [1]
[0 0] [1 1] [0 0] [0 0] [1]
h(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)
, f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)
, g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)}
Weak Trs:
{ h(x, x) -> x
, g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4, x5) = [1 1] x1 + [1 1] x2 + [0 0] x3 + [1 1] x4 + [0 0] x5 + [1]
[0 0] [0 0] [1 1] [0 0] [1 1] [1]
a() = [0]
[0]
g1(x1, x2, x3, x4) = [1 1] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [1]
[0 0] [1 1] [0 0] [0 0] [1]
g2(x1, x2, x3, x4) = [1 0] x1 + [0 1] x2 + [1 0] x3 + [0 0] x4 + [1]
[0 1] [1 0] [0 1] [0 0] [1]
h(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)
, f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)}
Weak Trs:
{ g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)
, h(x, x) -> x
, g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4, x5) = [1 2] x1 + [1 0] x2 + [1 1] x3 + [0 0] x4 + [1 0] x5 + [1]
[0 0] [0 1] [0 0] [0 1] [0 0] [1]
a() = [0]
[0]
g1(x1, x2, x3, x4) = [1 0] x1 + [0 1] x2 + [1 0] x3 + [0 0] x4 + [1]
[0 1] [0 0] [0 1] [0 0] [1]
g2(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [0]
[0 0] [0 1] [0 0] [0 0] [1]
h(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)}
Weak Trs:
{ f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)
, g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)
, h(x, x) -> x
, g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4, x5) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0 0] x4 + [1 0] x5 + [1]
[1 1] [1 1] [1 1] [0 0] [1 1] [1]
a() = [0]
[0]
g1(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [0]
[1 1] [0 0] [1 1] [0 0] [1]
g2(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [1]
[1 1] [0 0] [1 1] [0 0] [1]
h(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [1 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)
, f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)
, g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)
, h(x, x) -> x
, g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ f(x, y, w, w, a()) -> g1(x, x, y, w)
, f(x, y, w, a(), a()) -> g1(y, x, x, w)
, f(x, y, a(), a(), w) -> g2(x, y, y, w)
, f(x, y, a(), w, w) -> g2(y, y, x, w)
, g1(x, x, y, a()) -> h(x, y)
, g1(y, x, x, a()) -> h(x, y)
, h(x, x) -> x
, g2(x, y, y, a()) -> h(x, y)
, g2(y, y, x, a()) -> h(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))