We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(0(), Z) -> rnil()
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, pi(X) -> 2ndspos(X, from(0()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, square(X) -> times(X, X)
, from(X) -> n__from(X)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(0(), Z) -> rnil()
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, pi(X) -> 2ndspos(X, from(0()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, square(X) -> times(X, X)
, from(X) -> n__from(X)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
2ndspos(x1, x2) = [0 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
rcons(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [1]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(0(), Z) -> rnil()
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, pi(X) -> 2ndspos(X, from(0()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, times(s(X), Y) -> plus(Y, times(X, Y))
, square(X) -> times(X, X)
, from(X) -> n__from(X)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
Weak Trs:
{ 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {pi(X) -> 2ndspos(X, from(0()))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
2ndspos(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
rcons(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [2]
[0 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [1]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(0(), Z) -> rnil()
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, times(s(X), Y) -> plus(Y, times(X, Y))
, square(X) -> times(X, X)
, from(X) -> n__from(X)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
Weak Trs:
{ pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {square(X) -> times(X, X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
2ndspos(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
rcons(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [1]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
square(x1) = [0 0] x1 + [2]
[1 1] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(0(), Z) -> rnil()
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, times(s(X), Y) -> plus(Y, times(X, Y))
, from(X) -> n__from(X)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
Weak Trs:
{ square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {plus(0(), Y) -> Y}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
2ndspos(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
rcons(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
square(x1) = [0 0] x1 + [0]
[0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(0(), Z) -> rnil()
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, plus(s(X), Y) -> s(plus(X, Y))
, times(s(X), Y) -> plus(Y, times(X, Y))
, from(X) -> n__from(X)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
Weak Trs:
{ plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {2ndsneg(0(), Z) -> rnil()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [0]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
2ndspos(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [3]
rcons(x1, x2) = [0 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [1]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [1]
square(x1) = [0 0] x1 + [0]
[0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, plus(s(X), Y) -> s(plus(X, Y))
, times(s(X), Y) -> plus(Y, times(X, Y))
, from(X) -> n__from(X)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
Weak Trs:
{ 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(n__from(X)) -> from(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
2ndspos(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [2]
[0 0] [3]
rcons(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [2]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [1]
square(x1) = [0 0] x1 + [0]
[0 1] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, plus(s(X), Y) -> s(plus(X, Y))
, times(s(X), Y) -> plus(Y, times(X, Y))
, from(X) -> n__from(X)
, activate(X) -> X}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {times(s(X), Y) -> plus(Y, times(X, Y))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [0]
n__from(x1) = [0 1] x1 + [0]
[1 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
2ndspos(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
rcons(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
times(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
square(x1) = [1 1] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, plus(s(X), Y) -> s(plus(X, Y))
, from(X) -> n__from(X)
, activate(X) -> X}
Weak Trs:
{ times(s(X), Y) -> plus(Y, times(X, Y))
, activate(n__from(X)) -> from(X)
, 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [3]
n__from(x1) = [0 1] x1 + [0]
[1 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
2ndspos(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 1] x2 + [3]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
rcons(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
posrecip(x1) = [0 1] x1 + [1]
[1 0] [0]
2ndsneg(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [0]
negrecip(x1) = [1 0] x1 + [0]
[0 0] [0]
pi(x1) = [1 1] x1 + [0]
[0 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [0]
times(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
square(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:
{ from(X) -> cons(X, n__from(s(X)))
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, plus(s(X), Y) -> s(plus(X, Y))
, from(X) -> n__from(X)
, activate(X) -> X}
Weak Trs:
{ 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, times(s(X), Y) -> plus(Y, times(X, Y))
, activate(n__from(X)) -> from(X)
, 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [0]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
s(x1) = [1 0] x1 + [3]
[0 1] [2]
2ndspos(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
rcons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
times(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
square(x1) = [1 1] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, plus(s(X), Y) -> s(plus(X, Y))
, from(X) -> n__from(X)
, activate(X) -> X}
Weak Trs:
{ 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, times(s(X), Y) -> plus(Y, times(X, Y))
, activate(n__from(X)) -> from(X)
, 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [3]
[1 0] [0 0] [3]
n__from(x1) = [0 0] x1 + [2]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
2ndspos(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [3]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
rcons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[1 1] [0 0] [1]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [1 1] [1]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
square(x1) = [0 0] x1 + [2]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, plus(s(X), Y) -> s(plus(X, Y))
, from(X) -> n__from(X)}
Weak Trs:
{ activate(X) -> X
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, times(s(X), Y) -> plus(Y, times(X, Y))
, activate(n__from(X)) -> from(X)
, 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {from(X) -> n__from(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [2]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [3]
[1 0] [0 0] [3]
n__from(x1) = [0 0] x1 + [1]
[0 0] [0]
s(x1) = [1 0] x1 + [3]
[0 1] [1]
2ndspos(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[1 0] [1 1] [0]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [2]
activate(x1) = [1 0] x1 + [2]
[0 1] [1]
rcons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
posrecip(x1) = [0 0] x1 + [2]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[1 0] [0 0] [0]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [3]
[1 0] [2]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
times(x1, x2) = [1 0] x1 + [0 0] x2 + [2]
[1 1] [0 0] [1]
square(x1) = [1 0] x1 + [2]
[1 1] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, plus(s(X), Y) -> s(plus(X, Y))}
Weak Trs:
{ from(X) -> n__from(X)
, activate(X) -> X
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, times(s(X), Y) -> plus(Y, times(X, Y))
, activate(n__from(X)) -> from(X)
, 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {from(X) -> cons(X, n__from(s(X)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [1]
[0 0] [1]
cons(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
n__from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
2ndspos(x1, x2) = [0 2] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [2]
rcons(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
posrecip(x1) = [0 0] x1 + [1]
[0 0] [0]
2ndsneg(x1, x2) = [0 2] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 2] x1 + [2]
[0 0] [1]
plus(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
times(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
square(x1) = [1 1] x1 + [0]
[0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {plus(s(X), Y) -> s(plus(X, Y))}
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, times(s(X), Y) -> plus(Y, times(X, Y))
, activate(n__from(X)) -> from(X)
, 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
We consider the following Problem:
Strict Trs: {plus(s(X), Y) -> s(plus(X, Y))}
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, activate(Z)))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
, times(s(X), Y) -> plus(Y, times(X, Y))
, activate(n__from(X)) -> from(X)
, 2ndsneg(0(), Z) -> rnil()
, plus(0(), Y) -> Y
, square(X) -> times(X, X)
, pi(X) -> 2ndspos(X, from(0()))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
, times(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
Uargs(s) = {1}, Uargs(2ndspos) = {2}, Uargs(cons2) = {2},
Uargs(activate) = {}, Uargs(rcons) = {2}, Uargs(posrecip) = {},
Uargs(2ndsneg) = {2}, Uargs(negrecip) = {}, Uargs(pi) = {},
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(square) = {}
We have the following restricted polynomial interpretation:
Interpretation Functions:
[from](x1) = 0
[cons](x1, x2) = x2
[n__from](x1) = 0
[s](x1) = 2 + x1
[2ndspos](x1, x2) = 2*x2
[0]() = 0
[rnil]() = 0
[cons2](x1, x2) = x2
[activate](x1) = x1
[rcons](x1, x2) = x2
[posrecip](x1) = 0
[2ndsneg](x1, x2) = 2*x2
[negrecip](x1) = 0
[pi](x1) = 0
[plus](x1, x2) = 2*x1 + x2
[times](x1, x2) = 2*x1 + x1*x2
[square](x1) = 2*x1 + 2*x1^2
Hurray, we answered YES(?,O(n^2))