We consider the following Problem:
Strict Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, filter(cons(X), s(N), M) -> cons(X)
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0()))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, filter(cons(X), s(N), M) -> cons(X)
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0()))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {zprimes() -> sieve(nats(s(s(0()))))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {1}, Uargs(nats) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
cons(x1) = [0 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve(x1) = [1 0] x1 + [0]
[0 0] [1]
nats(x1) = [0 0] x1 + [0]
[0 0] [0]
zprimes() = [2]
[2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, filter(cons(X), s(N), M) -> cons(X)
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, nats(N) -> cons(N)}
Weak Trs: {zprimes() -> sieve(nats(s(s(0()))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {nats(N) -> cons(N)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {1}, Uargs(nats) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
cons(x1) = [0 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve(x1) = [1 0] x1 + [0]
[0 0] [1]
nats(x1) = [0 0] x1 + [2]
[0 0] [2]
zprimes() = [2]
[2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, filter(cons(X), s(N), M) -> cons(X)
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))}
Weak Trs:
{ nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0()))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {filter(cons(X), s(N), M) -> cons(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {1}, Uargs(nats) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[0 0] [0 1] [0 0] [1]
cons(x1) = [0 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
s(x1) = [0 0] x1 + [1]
[0 0] [0]
sieve(x1) = [1 0] x1 + [0]
[0 1] [0]
nats(x1) = [0 0] x1 + [1]
[0 0] [1]
zprimes() = [2]
[2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))}
Weak Trs:
{ filter(cons(X), s(N), M) -> cons(X)
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0()))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {1}, Uargs(nats) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [1 0] [1 0] [1]
cons(x1) = [0 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve(x1) = [1 0] x1 + [2]
[1 0] [0]
nats(x1) = [0 0] x1 + [1]
[0 0] [2]
zprimes() = [3]
[2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {filter(cons(X), 0(), M) -> cons(0())}
Weak Trs:
{ sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, filter(cons(X), s(N), M) -> cons(X)
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0()))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {filter(cons(X), 0(), M) -> cons(0())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {1}, Uargs(nats) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
cons(x1) = [0 0] x1 + [0]
[0 0] [1]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve(x1) = [1 0] x1 + [1]
[0 0] [1]
nats(x1) = [0 0] x1 + [0]
[0 0] [2]
zprimes() = [2]
[2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, filter(cons(X), s(N), M) -> cons(X)
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0()))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ filter(cons(X), 0(), M) -> cons(0())
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, filter(cons(X), s(N), M) -> cons(X)
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0()))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))