We consider the following Problem:
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, sqr(0()) -> 0()
, sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
, add(0(), X) -> X
, add(s(X), Y) -> s(n__add(activate(X), Y))
, first(0(), X) -> nil()
, first(s(X), cons(Y, Z)) ->
cons(Y, n__first(activate(X), activate(Z)))
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, s(X) -> n__s(X)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__terms(X)) -> terms(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(n__s(X)) -> s(X)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
Arguments of following rules are not normal-forms:
{ sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
, dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
, add(s(X), Y) -> s(n__add(activate(X), Y))
, first(s(X), cons(Y, Z)) ->
cons(Y, n__first(activate(X), activate(Z)))}
All above mentioned rules can be savely removed.
We consider the following Problem:
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, sqr(0()) -> 0()
, dbl(0()) -> 0()
, add(0(), X) -> X
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, s(X) -> n__s(X)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__terms(X)) -> terms(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(n__s(X)) -> s(X)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, activate(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:
{ terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {1, 2}, Uargs(recip) = {1},
Uargs(sqr) = {}, Uargs(n__terms) = {1}, Uargs(s) = {},
Uargs(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
Uargs(first) = {}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [2]
[1 0] [2]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
n__add(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[1 0] [1]
dbl(x1) = [0 0] x1 + [1]
[0 0] [1]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sqr(0()) -> 0()
, add(0(), X) -> X
, s(X) -> n__s(X)
, activate(n__terms(X)) -> terms(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, activate(X) -> X}
Weak Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {sqr(0()) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {1, 2}, Uargs(recip) = {1},
Uargs(sqr) = {}, Uargs(n__terms) = {1}, Uargs(s) = {},
Uargs(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
Uargs(first) = {}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [2]
[1 0] [2]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [1]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
n__add(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[1 0] [1]
dbl(x1) = [0 0] x1 + [1]
[0 0] [1]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ add(0(), X) -> X
, s(X) -> n__s(X)
, activate(n__terms(X)) -> terms(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, activate(X) -> X}
Weak Trs:
{ sqr(0()) -> 0()
, terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {add(0(), X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {1, 2}, Uargs(recip) = {1},
Uargs(sqr) = {}, Uargs(n__terms) = {1}, Uargs(s) = {},
Uargs(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
Uargs(first) = {}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [2]
[0 1] [2]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[1 0] [0 0] [0]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [1]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
n__add(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[1 0] [1]
dbl(x1) = [0 0] x1 + [1]
[0 0] [1]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ s(X) -> n__s(X)
, activate(n__terms(X)) -> terms(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, activate(X) -> X}
Weak Trs:
{ add(0(), X) -> X
, sqr(0()) -> 0()
, terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(n__first(X1, X2)) -> first(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {1, 2}, Uargs(recip) = {1},
Uargs(sqr) = {}, Uargs(n__terms) = {1}, Uargs(s) = {},
Uargs(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
Uargs(first) = {}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [2]
[0 1] [2]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[1 0] [0 0] [0]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [1]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
n__add(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[1 0] [1]
dbl(x1) = [0 0] x1 + [1]
[0 0] [1]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ s(X) -> n__s(X)
, activate(n__terms(X)) -> terms(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(n__dbl(X)) -> dbl(X)
, activate(X) -> X}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, add(0(), X) -> X
, sqr(0()) -> 0()
, terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ activate(n__terms(X)) -> terms(X)
, activate(n__dbl(X)) -> dbl(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {1, 2}, Uargs(recip) = {1},
Uargs(sqr) = {}, Uargs(n__terms) = {1}, Uargs(s) = {},
Uargs(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
Uargs(first) = {}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [2]
[1 0] [2]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [1]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [1]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
n__add(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [2]
[1 0] [1]
dbl(x1) = [0 0] x1 + [0]
[0 0] [1]
n__s(x1) = [0 0] x1 + [3]
[0 0] [0]
n__dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ s(X) -> n__s(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(X) -> X}
Weak Trs:
{ activate(n__terms(X)) -> terms(X)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, add(0(), X) -> X
, sqr(0()) -> 0()
, terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
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(terms) = {}, Uargs(cons) = {1, 2}, Uargs(recip) = {1},
Uargs(sqr) = {}, Uargs(n__terms) = {1}, Uargs(s) = {},
Uargs(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
Uargs(first) = {}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [1]
[0 1] [1]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [1 0] x1 + [0]
[0 1] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 1] [1]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
n__add(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [1]
activate(x1) = [1 0] x1 + [1]
[0 1] [3]
dbl(x1) = [0 0] x1 + [1]
[1 1] [1]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__dbl(x1) = [0 0] x1 + [0]
[1 1] [0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [1]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[1 1] [1 1] [1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[1 1] [1 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ s(X) -> n__s(X)
, activate(n__add(X1, X2)) -> add(X1, X2)}
Weak Trs:
{ activate(X) -> X
, activate(n__terms(X)) -> terms(X)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, add(0(), X) -> X
, sqr(0()) -> 0()
, terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(n__add(X1, X2)) -> add(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {1, 2}, Uargs(recip) = {1},
Uargs(sqr) = {}, Uargs(n__terms) = {1}, Uargs(s) = {},
Uargs(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
Uargs(first) = {}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [1]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
0() = [0]
[1]
n__add(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
dbl(x1) = [0 0] x1 + [1]
[0 0] [1]
n__s(x1) = [1 0] x1 + [0]
[0 0] [0]
n__dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {s(X) -> n__s(X)}
Weak Trs:
{ activate(n__add(X1, X2)) -> add(X1, X2)
, activate(X) -> X
, activate(n__terms(X)) -> terms(X)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, add(0(), X) -> X
, sqr(0()) -> 0()
, terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {s(X) -> n__s(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {}, Uargs(cons) = {1, 2}, Uargs(recip) = {1},
Uargs(sqr) = {}, Uargs(n__terms) = {1}, Uargs(s) = {},
Uargs(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
Uargs(first) = {}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [1]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [0 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
0() = [0]
[0]
n__add(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
dbl(x1) = [0 0] x1 + [1]
[0 0] [1]
n__s(x1) = [1 0] x1 + [0]
[0 0] [0]
n__dbl(x1) = [0 0] x1 + [0]
[0 0] [0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ s(X) -> n__s(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(X) -> X
, activate(n__terms(X)) -> terms(X)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, add(0(), X) -> X
, sqr(0()) -> 0()
, terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ s(X) -> n__s(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(X) -> X
, activate(n__terms(X)) -> terms(X)
, activate(n__dbl(X)) -> dbl(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, add(0(), X) -> X
, sqr(0()) -> 0()
, terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))