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