We consider the following Problem:
Strict Trs:
{ a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(X) -> f(X)
, a__p(X) -> p(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(X) -> f(X)
, a__p(X) -> p(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ mark(0()) -> 0()
, a__f(X) -> f(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(cons) = {1}, Uargs(f) = {},
Uargs(s) = {1}, Uargs(a__p) = {1}, Uargs(mark) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
a__p(x1) = [1 0] x1 + [0]
[0 0] [1]
mark(x1) = [0 0] x1 + [1]
[0 0] [1]
p(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:
{ a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)}
Weak Trs:
{ mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__p(s(0())) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(cons) = {1}, Uargs(f) = {},
Uargs(s) = {1}, Uargs(a__p) = {1}, Uargs(mark) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [2]
a__p(x1) = [1 2] x1 + [0]
[0 0] [0]
mark(x1) = [0 0] x1 + [0]
[0 0] [3]
p(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:
{ a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)}
Weak Trs:
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__p(X) -> p(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(cons) = {1}, Uargs(f) = {},
Uargs(s) = {1}, Uargs(a__p) = {1}, Uargs(mark) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 1] x1 + [1]
[0 0] [1]
0() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
a__p(x1) = [1 0] x1 + [2]
[0 1] [2]
mark(x1) = [0 0] x1 + [2]
[0 0] [3]
p(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:
{ a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ a__p(X) -> p(X)
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__f(0()) -> cons(0(), f(s(0())))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(cons) = {1}, Uargs(f) = {},
Uargs(s) = {1}, Uargs(a__p) = {1}, Uargs(mark) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
a__p(x1) = [1 0] x1 + [2]
[0 0] [1]
mark(x1) = [0 0] x1 + [3]
[0 0] [3]
p(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:
{ a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ a__f(0()) -> cons(0(), f(s(0())))
, a__p(X) -> p(X)
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__f(s(0())) -> a__f(a__p(s(0())))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(cons) = {1}, Uargs(f) = {},
Uargs(s) = {1}, Uargs(a__p) = {1}, Uargs(mark) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [1]
[1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [3]
a__p(x1) = [1 0] x1 + [1]
[0 0] [2]
mark(x1) = [1 0] x1 + [0]
[0 0] [1]
p(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(0()) -> cons(0(), f(s(0())))
, a__p(X) -> p(X)
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(f(X)) -> a__f(mark(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(cons) = {1}, Uargs(f) = {},
Uargs(s) = {1}, Uargs(a__p) = {1}, Uargs(mark) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [0]
[0 1] [1]
0() = [0]
[1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 1] [1]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
a__p(x1) = [1 0] x1 + [0]
[0 1] [0]
mark(x1) = [0 1] x1 + [0]
[0 1] [0]
p(x1) = [0 0] x1 + [0]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ mark(f(X)) -> a__f(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(0()) -> cons(0(), f(s(0())))
, a__p(X) -> p(X)
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(cons(X1, X2)) -> cons(mark(X1), X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(cons) = {1}, Uargs(f) = {},
Uargs(s) = {1}, Uargs(a__p) = {1}, Uargs(mark) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [0]
[0 1] [1]
0() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [1]
f(x1) = [0 0] x1 + [0]
[0 1] [1]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
a__p(x1) = [1 0] x1 + [0]
[0 1] [0]
mark(x1) = [0 2] x1 + [0]
[0 1] [0]
p(x1) = [0 0] x1 + [0]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(f(X)) -> a__f(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(0()) -> cons(0(), f(s(0())))
, a__p(X) -> p(X)
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(s(X)) -> s(mark(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(cons) = {1}, Uargs(f) = {},
Uargs(s) = {1}, Uargs(a__p) = {1}, Uargs(mark) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [0]
[0 1] [0]
0() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 1] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
a__p(x1) = [1 0] x1 + [0]
[0 1] [0]
mark(x1) = [0 1] x1 + [0]
[0 1] [0]
p(x1) = [0 0] x1 + [0]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {mark(p(X)) -> a__p(mark(X))}
Weak Trs:
{ mark(s(X)) -> s(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(f(X)) -> a__f(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(0()) -> cons(0(), f(s(0())))
, a__p(X) -> p(X)
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {mark(p(X)) -> a__p(mark(X))}
Weak Trs:
{ mark(s(X)) -> s(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(f(X)) -> a__f(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(0()) -> cons(0(), f(s(0())))
, a__p(X) -> p(X)
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_0(1) -> 1
, a__f_0(2) -> 1
, a__f_1(3) -> 3
, a__f_1(4) -> 1
, 0_0() -> 1
, 0_0() -> 2
, 0_1() -> 1
, 0_1() -> 3
, 0_1() -> 4
, 0_1() -> 6
, cons_0(1, 2) -> 1
, cons_0(2, 2) -> 1
, cons_0(2, 2) -> 2
, cons_1(3, 2) -> 3
, cons_1(6, 7) -> 1
, cons_1(6, 7) -> 3
, f_0(1) -> 1
, f_0(2) -> 1
, f_0(2) -> 2
, f_1(3) -> 3
, f_1(4) -> 1
, f_1(5) -> 7
, s_0(1) -> 1
, s_0(2) -> 2
, s_1(3) -> 3
, s_1(6) -> 5
, a__p_0(2) -> 1
, a__p_1(3) -> 1
, a__p_1(3) -> 3
, a__p_1(5) -> 4
, mark_0(2) -> 1
, mark_1(2) -> 3
, p_0(2) -> 1
, p_0(2) -> 2
, p_1(3) -> 1
, p_1(3) -> 3
, p_1(5) -> 4}
Hurray, we answered YES(?,O(n^1))