We consider the following Problem:
Strict Trs:
{ f(X) -> if(X, c(), n__f(true()))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> activate(Y)
, f(X) -> n__f(X)
, activate(n__f(X)) -> f(X)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(X) -> if(X, c(), n__f(true()))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> activate(Y)
, f(X) -> n__f(X)
, activate(n__f(X)) -> f(X)
, 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:
{ f(X) -> if(X, c(), n__f(true()))
, f(X) -> n__f(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(if) = {}, Uargs(n__f) = {},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [2]
[0 0] [2]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
c() = [0]
[0]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
activate(x1) = [1 0] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ if(true(), X, Y) -> X
, if(false(), X, Y) -> activate(Y)
, activate(n__f(X)) -> f(X)
, activate(X) -> X}
Weak Trs:
{ f(X) -> if(X, c(), n__f(true()))
, f(X) -> n__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: {if(true(), X, Y) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(if) = {}, Uargs(n__f) = {},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [2]
[0 0] [2]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[0 0] [0 1] [0 0] [1]
c() = [0]
[1]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
activate(x1) = [1 0] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ if(false(), X, Y) -> activate(Y)
, activate(n__f(X)) -> f(X)
, activate(X) -> X}
Weak Trs:
{ if(true(), X, Y) -> X
, f(X) -> if(X, c(), n__f(true()))
, f(X) -> n__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: {activate(n__f(X)) -> f(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(if) = {}, Uargs(n__f) = {},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[0 0] [1]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[0 0] [0 1] [0 0] [1]
c() = [0]
[0]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
activate(x1) = [1 0] x1 + [3]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ if(false(), X, Y) -> activate(Y)
, activate(X) -> X}
Weak Trs:
{ activate(n__f(X)) -> f(X)
, if(true(), X, Y) -> X
, f(X) -> if(X, c(), n__f(true()))
, f(X) -> n__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: {if(false(), X, Y) -> activate(Y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(if) = {}, Uargs(n__f) = {},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [1 0] x1 + [0]
[0 0] [1]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 1] [0 0] [1]
c() = [0]
[0]
n__f(x1) = [1 0] x1 + [0]
[0 0] [0]
true() = [0]
[0]
false() = [1]
[0]
activate(x1) = [1 0] x1 + [0]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {activate(X) -> X}
Weak Trs:
{ if(false(), X, Y) -> activate(Y)
, activate(n__f(X)) -> f(X)
, if(true(), X, Y) -> X
, f(X) -> if(X, c(), n__f(true()))
, f(X) -> n__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: {activate(X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(if) = {}, Uargs(n__f) = {},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[0 0] [1]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [0 1] [0 1] [1]
c() = [0]
[0]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ activate(X) -> X
, if(false(), X, Y) -> activate(Y)
, activate(n__f(X)) -> f(X)
, if(true(), X, Y) -> X
, f(X) -> if(X, c(), n__f(true()))
, f(X) -> n__f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(X) -> X
, if(false(), X, Y) -> activate(Y)
, activate(n__f(X)) -> f(X)
, if(true(), X, Y) -> X
, f(X) -> if(X, c(), n__f(true()))
, f(X) -> n__f(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))