We consider the following Problem:
Strict Trs:
{ not(x) -> xor(x, true())
, implies(x, y) -> xor(and(x, y), xor(x, true()))
, or(x, y) -> xor(and(x, y), xor(x, y))
, =(x, y) -> xor(x, xor(y, true()))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ not(x) -> xor(x, true())
, implies(x, y) -> xor(and(x, y), xor(x, true()))
, or(x, y) -> xor(and(x, y), xor(x, y))
, =(x, y) -> xor(x, xor(y, true()))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {not(x) -> xor(x, true())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(xor) = {}, Uargs(implies) = {},
Uargs(and) = {}, Uargs(or) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [2]
[0 0] [2]
xor(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
implies(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
or(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
=(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:
{ implies(x, y) -> xor(and(x, y), xor(x, true()))
, or(x, y) -> xor(and(x, y), xor(x, y))
, =(x, y) -> xor(x, xor(y, true()))}
Weak Trs: {not(x) -> xor(x, true())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {implies(x, y) -> xor(and(x, y), xor(x, true()))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(xor) = {}, Uargs(implies) = {},
Uargs(and) = {}, Uargs(or) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [2]
[0 0] [2]
xor(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
implies(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
or(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
=(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:
{ or(x, y) -> xor(and(x, y), xor(x, y))
, =(x, y) -> xor(x, xor(y, true()))}
Weak Trs:
{ implies(x, y) -> xor(and(x, y), xor(x, true()))
, not(x) -> xor(x, true())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {or(x, y) -> xor(and(x, y), xor(x, y))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(xor) = {}, Uargs(implies) = {},
Uargs(and) = {}, Uargs(or) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [2]
[0 0] [2]
xor(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
implies(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
or(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
=(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: {=(x, y) -> xor(x, xor(y, true()))}
Weak Trs:
{ or(x, y) -> xor(and(x, y), xor(x, y))
, implies(x, y) -> xor(and(x, y), xor(x, true()))
, not(x) -> xor(x, true())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {=(x, y) -> xor(x, xor(y, true()))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(xor) = {}, Uargs(implies) = {},
Uargs(and) = {}, Uargs(or) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [2]
[0 0] [2]
xor(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
implies(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
or(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
=(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ =(x, y) -> xor(x, xor(y, true()))
, or(x, y) -> xor(and(x, y), xor(x, y))
, implies(x, y) -> xor(and(x, y), xor(x, true()))
, not(x) -> xor(x, true())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ =(x, y) -> xor(x, xor(y, true()))
, or(x, y) -> xor(and(x, y), xor(x, y))
, implies(x, y) -> xor(and(x, y), xor(x, true()))
, not(x) -> xor(x, true())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))