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