We consider the following Problem:
Strict Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, i(f(x, h(y))) -> y
, i(h2(s(x), y, h1(x, z))) -> z
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(x), y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, i(f(x, h(y))) -> y
, i(h2(s(x), y, h1(x, z))) -> z
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(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:
{ k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(x), y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {2}, Uargs(j) = {}, Uargs(g) = {1}, Uargs(k) = {},
Uargs(h1) = {}, Uargs(h2) = {}, Uargs(s) = {}, Uargs(i) = {},
Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[1 0] [1 0] [1]
j(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 0] x1 + [0]
[1 0] [0]
k(x1) = [0 0] x1 + [1]
[0 0] [1]
h1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[1 0] [0 0] [0]
h2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 1] [1]
0() = [0]
[0]
s(x1) = [0 0] x1 + [1]
[0 0] [0]
i(x1) = [0 1] x1 + [0]
[0 1] [0]
h(x1) = [0 1] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, i(f(x, h(y))) -> y
, i(h2(s(x), y, h1(x, z))) -> z}
Weak Trs:
{ k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(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: {i(f(x, h(y))) -> y}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {2}, Uargs(j) = {}, Uargs(g) = {1}, Uargs(k) = {},
Uargs(h1) = {}, Uargs(h2) = {}, Uargs(s) = {}, Uargs(i) = {},
Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[1 0] [0 1] [0]
j(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [0]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
k(x1) = [1 1] x1 + [1]
[0 0] [3]
h1(x1, x2) = [0 1] x1 + [1 0] x2 + [2]
[0 0] [0 0] [2]
h2(x1, x2, x3) = [0 1] x1 + [0 0] x2 + [1 1] x3 + [0]
[1 0] [0 0] [0 0] [1]
0() = [0]
[1]
s(x1) = [1 0] x1 + [0]
[0 0] [2]
i(x1) = [0 1] x1 + [1]
[1 0] [1]
h(x1) = [0 1] x1 + [2]
[1 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, i(h2(s(x), y, h1(x, z))) -> z}
Weak Trs:
{ i(f(x, h(y))) -> y
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(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, h1(y, z)) -> h2(0(), x, h1(y, z))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {2}, Uargs(j) = {}, Uargs(g) = {1}, Uargs(k) = {},
Uargs(h1) = {}, Uargs(h2) = {}, Uargs(s) = {}, Uargs(i) = {},
Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
j(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[0 0] [1 0] [0]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
k(x1) = [0 0] x1 + [0]
[0 1] [1]
h1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[1 0] [1 0] [0]
h2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 1] [1]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
i(x1) = [0 1] x1 + [0]
[1 0] [1]
h(x1) = [0 1] x1 + [0]
[1 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, i(h2(s(x), y, h1(x, z))) -> z}
Weak Trs:
{ f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, i(f(x, h(y))) -> y
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(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: {g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {2}, Uargs(j) = {}, Uargs(g) = {1}, Uargs(k) = {},
Uargs(h1) = {}, Uargs(h2) = {}, Uargs(s) = {}, Uargs(i) = {},
Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
j(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 0] x1 + [2]
[0 0] [1]
k(x1) = [0 0] x1 + [0]
[0 0] [1]
h1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
h2(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
i(x1) = [1 0] x1 + [0]
[0 1] [0]
h(x1) = [1 0] x1 + [0]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, i(h2(s(x), y, h1(x, z))) -> z}
Weak Trs:
{ g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, i(f(x, h(y))) -> y
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(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:
{h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {2}, Uargs(j) = {}, Uargs(g) = {1}, Uargs(k) = {},
Uargs(h1) = {}, Uargs(h2) = {}, Uargs(s) = {}, Uargs(i) = {},
Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
j(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 0] x1 + [1]
[0 0] [1]
k(x1) = [0 0] x1 + [3]
[0 0] [1]
h1(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
h2(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
i(x1) = [1 0] x1 + [1]
[0 1] [1]
h(x1) = [1 0] x1 + [1]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, i(h2(s(x), y, h1(x, z))) -> z}
Weak Trs:
{ h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, i(f(x, h(y))) -> y
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(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: {i(h2(s(x), y, h1(x, z))) -> z}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {2}, Uargs(j) = {}, Uargs(g) = {1}, Uargs(k) = {},
Uargs(h1) = {}, Uargs(h2) = {}, Uargs(s) = {}, Uargs(i) = {},
Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
j(x1, x2) = [1 0] x1 + [0 0] x2 + [3]
[0 0] [0 0] [0]
g(x1) = [1 0] x1 + [0]
[0 1] [1]
k(x1) = [1 0] x1 + [2]
[0 1] [0]
h1(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [0]
h2(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 1] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [2]
[0 0] [0]
i(x1) = [1 0] x1 + [1]
[0 1] [1]
h(x1) = [1 0] x1 + [0]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {f(j(x, y), y) -> g(f(x, k(y)))}
Weak Trs:
{ i(h2(s(x), y, h1(x, z))) -> z
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, i(f(x, h(y))) -> y
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(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(j(x, y), y) -> g(f(x, k(y)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {2}, Uargs(j) = {}, Uargs(g) = {1}, Uargs(k) = {},
Uargs(h1) = {}, Uargs(h2) = {}, Uargs(s) = {}, Uargs(i) = {},
Uargs(h) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
j(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [2]
g(x1) = [1 0] x1 + [1]
[0 1] [0]
k(x1) = [1 0] x1 + [0]
[0 1] [0]
h1(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
h2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 1] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
i(x1) = [1 0] x1 + [1]
[0 1] [1]
h(x1) = [1 0] x1 + [0]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, i(h2(s(x), y, h1(x, z))) -> z
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, i(f(x, h(y))) -> y
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(x), y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ f(j(x, y), y) -> g(f(x, k(y)))
, i(h2(s(x), y, h1(x, z))) -> z
, h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
, g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
, f(x, h1(y, z)) -> h2(0(), x, h1(y, z))
, i(f(x, h(y))) -> y
, k(h(x)) -> h1(0(), x)
, k(h1(x, y)) -> h1(s(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))