We consider the following Problem:
Strict Trs:
{ a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__length1(X) -> a__length(X)
, mark(from(X)) -> a__from(mark(X))
, mark(length(X)) -> a__length(X)
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(nil()) -> nil()
, mark(0()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)
, a__length1(X) -> length1(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
We consider the following Problem:
Strict Trs:
{ a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__length1(X) -> a__length(X)
, mark(from(X)) -> a__from(mark(X))
, mark(length(X)) -> a__length(X)
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(nil()) -> nil()
, mark(0()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)
, a__length1(X) -> length1(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 0] x1 + [1]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
mark(x1) = [0 0] x1 + [0]
[0 0] [1]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
a__length(x1) = [0 0] x1 + [1]
[0 0] [1]
nil() = [0]
[0]
0() = [0]
[0]
a__length1(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
length1(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__from(X) -> cons(mark(X), from(s(X)))
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__length1(X) -> a__length(X)
, mark(from(X)) -> a__from(mark(X))
, mark(length(X)) -> a__length(X)
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(nil()) -> nil()
, mark(0()) -> 0()
, a__length1(X) -> length1(X)}
Weak Trs:
{ a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__from(X) -> cons(mark(X), from(s(X)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 0] x1 + [3]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
mark(x1) = [0 0] x1 + [0]
[0 0] [1]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
a__length(x1) = [0 0] x1 + [1]
[0 0] [1]
nil() = [0]
[0]
0() = [0]
[0]
a__length1(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
length1(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__length(cons(X, Y)) -> s(a__length1(Y))
, a__length1(X) -> a__length(X)
, mark(from(X)) -> a__from(mark(X))
, mark(length(X)) -> a__length(X)
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(nil()) -> nil()
, mark(0()) -> 0()
, a__length1(X) -> length1(X)}
Weak Trs:
{ a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__length(cons(X, Y)) -> s(a__length1(Y))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 0] x1 + [1]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
mark(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
a__length(x1) = [0 0] x1 + [1]
[0 0] [1]
nil() = [0]
[0]
0() = [0]
[0]
a__length1(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
length1(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__length1(X) -> a__length(X)
, mark(from(X)) -> a__from(mark(X))
, mark(length(X)) -> a__length(X)
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(nil()) -> nil()
, mark(0()) -> 0()
, a__length1(X) -> length1(X)}
Weak Trs:
{ a__length(cons(X, Y)) -> s(a__length1(Y))
, a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__length1(X) -> length1(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 0] x1 + [1]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
mark(x1) = [0 0] x1 + [0]
[0 0] [1]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
a__length(x1) = [0 0] x1 + [3]
[0 0] [1]
nil() = [0]
[0]
0() = [0]
[0]
a__length1(x1) = [0 0] x1 + [1]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
length1(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__length1(X) -> a__length(X)
, mark(from(X)) -> a__from(mark(X))
, mark(length(X)) -> a__length(X)
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(nil()) -> nil()
, mark(0()) -> 0()}
Weak Trs:
{ a__length1(X) -> length1(X)
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__length1(X) -> a__length(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 0] x1 + [0]
[0 0] [3]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [3]
mark(x1) = [1 0] x1 + [0]
[0 0] [0]
from(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 1] x1 + [2]
[0 0] [1]
a__length(x1) = [0 3] x1 + [0]
[0 0] [1]
nil() = [0]
[2]
0() = [1]
[0]
a__length1(x1) = [0 3] x1 + [1]
[0 0] [1]
length(x1) = [0 3] x1 + [0]
[0 0] [0]
length1(x1) = [0 3] x1 + [1]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(from(X)) -> a__from(mark(X))
, mark(length(X)) -> a__length(X)
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(nil()) -> nil()
, mark(0()) -> 0()}
Weak Trs:
{ a__length1(X) -> a__length(X)
, a__length1(X) -> length1(X)
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(length(X)) -> a__length(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 2] x1 + [0]
[0 0] [2]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
mark(x1) = [1 2] x1 + [0]
[0 0] [2]
from(x1) = [1 2] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
a__length(x1) = [0 0] x1 + [0]
[0 0] [2]
nil() = [0]
[0]
0() = [0]
[0]
a__length1(x1) = [0 0] x1 + [0]
[0 0] [2]
length(x1) = [0 0] x1 + [0]
[0 0] [2]
length1(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:
{ mark(from(X)) -> a__from(mark(X))
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(nil()) -> nil()
, mark(0()) -> 0()}
Weak Trs:
{ mark(length(X)) -> a__length(X)
, a__length1(X) -> a__length(X)
, a__length1(X) -> length1(X)
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(nil()) -> nil()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 2] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 1] [0 1] [0]
mark(x1) = [1 2] x1 + [0]
[0 0] [1]
from(x1) = [1 2] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
a__length(x1) = [0 1] x1 + [0]
[0 0] [0]
nil() = [3]
[1]
0() = [0]
[0]
a__length1(x1) = [0 1] x1 + [0]
[0 0] [0]
length(x1) = [0 1] x1 + [0]
[0 0] [0]
length1(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:
{ mark(from(X)) -> a__from(mark(X))
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, mark(0()) -> 0()}
Weak Trs:
{ mark(nil()) -> nil()
, mark(length(X)) -> a__length(X)
, a__length1(X) -> a__length(X)
, a__length1(X) -> length1(X)
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(0()) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 2] x1 + [1]
[0 0] [3]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 1] [2]
mark(x1) = [1 2] x1 + [0]
[0 0] [1]
from(x1) = [1 2] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
a__length(x1) = [0 2] x1 + [0]
[0 0] [1]
nil() = [0]
[0]
0() = [0]
[1]
a__length1(x1) = [0 2] x1 + [0]
[0 0] [1]
length(x1) = [0 2] x1 + [0]
[0 0] [0]
length1(x1) = [0 2] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(from(X)) -> a__from(mark(X))
, mark(length1(X)) -> a__length1(X)
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ mark(0()) -> 0()
, mark(nil()) -> nil()
, mark(length(X)) -> a__length(X)
, a__length1(X) -> a__length(X)
, a__length1(X) -> length1(X)
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(length1(X)) -> a__length1(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 1] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
mark(x1) = [1 1] x1 + [0]
[0 0] [1]
from(x1) = [1 1] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
a__length(x1) = [0 0] x1 + [3]
[0 0] [1]
nil() = [1]
[0]
0() = [0]
[0]
a__length1(x1) = [0 0] x1 + [3]
[0 0] [1]
length(x1) = [0 0] x1 + [2]
[0 0] [1]
length1(x1) = [0 0] x1 + [3]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(from(X)) -> a__from(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ mark(length1(X)) -> a__length1(X)
, mark(0()) -> 0()
, mark(nil()) -> nil()
, mark(length(X)) -> a__length(X)
, a__length1(X) -> a__length(X)
, a__length1(X) -> length1(X)
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
We consider the following Problem:
Strict Trs:
{ mark(from(X)) -> a__from(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ mark(length1(X)) -> a__length1(X)
, mark(0()) -> 0()
, mark(nil()) -> nil()
, mark(length(X)) -> a__length(X)
, a__length1(X) -> a__length(X)
, a__length1(X) -> length1(X)
, a__length(cons(X, Y)) -> s(a__length1(Y))
, a__from(X) -> cons(mark(X), from(s(X)))
, a__length(nil()) -> 0()
, a__from(X) -> from(X)
, a__length(X) -> length(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The following argument positions are usable:
Uargs(a__from) = {1}, Uargs(cons) = {1}, Uargs(mark) = {},
Uargs(from) = {}, Uargs(s) = {1}, Uargs(a__length) = {},
Uargs(a__length1) = {}, Uargs(length) = {}, Uargs(length1) = {}
We have the following constructor-based EDA-non-satisfying and IDA(2)-non-satisfying matrix interpretation:
Interpretation Functions:
a__from(x1) = [1 2 0] x1 + [1]
[0 1 0] [2]
[0 0 0] [1]
cons(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 1] [1]
[0 0 0] [0 0 1] [1]
mark(x1) = [1 2 0] x1 + [0]
[0 1 0] [0]
[0 2 0] [0]
from(x1) = [1 2 0] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [1]
[0 0 0] [0]
a__length(x1) = [0 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
nil() = [1]
[2]
[2]
0() = [0]
[1]
[0]
a__length1(x1) = [0 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
length(x1) = [0 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
length1(x1) = [0 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
Hurray, we answered YES(?,O(n^2))