We consider the following Problem:
Strict Trs:
{ not(x) -> if(x, false(), true())
, and(x, y) -> if(x, y, false())
, or(x, y) -> if(x, true(), y)
, implies(x, y) -> if(x, y, true())
, =(x, x) -> true()
, =(x, y) -> if(x, y, not(y))
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, if(x, x, if(x, false(), true())) -> true()
, =(x, y) -> if(x, y, if(y, false(), true()))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ not(x) -> if(x, false(), true())
, and(x, y) -> if(x, y, false())
, or(x, y) -> if(x, true(), y)
, implies(x, y) -> if(x, y, true())
, =(x, x) -> true()
, =(x, y) -> if(x, y, not(y))
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, if(x, x, if(x, false(), true())) -> true()
, =(x, y) -> if(x, y, if(y, false(), true()))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> true()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(if) = {3}, Uargs(and) = {}, Uargs(or) = {},
Uargs(implies) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
false() = [0]
[0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
or(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
implies(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
[0 0] [0 0] [0]
=(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ not(x) -> if(x, false(), true())
, or(x, y) -> if(x, true(), y)
, implies(x, y) -> if(x, y, true())
, =(x, x) -> true()
, =(x, y) -> if(x, y, not(y))
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, =(x, y) -> if(x, y, if(y, false(), true()))}
Weak Trs:
{ and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> 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) -> if(x, true(), y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(if) = {3}, Uargs(and) = {}, Uargs(or) = {},
Uargs(implies) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
false() = [0]
[0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
or(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [2]
implies(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
[0 0] [0 0] [0]
=(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ not(x) -> if(x, false(), true())
, implies(x, y) -> if(x, y, true())
, =(x, x) -> true()
, =(x, y) -> if(x, y, not(y))
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, =(x, y) -> if(x, y, if(y, false(), true()))}
Weak Trs:
{ or(x, y) -> if(x, true(), y)
, and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> 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) -> if(x, y, true())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(if) = {3}, Uargs(and) = {}, Uargs(or) = {},
Uargs(implies) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
false() = [0]
[0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
or(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [2]
implies(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
=(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ not(x) -> if(x, false(), true())
, =(x, x) -> true()
, =(x, y) -> if(x, y, not(y))
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, =(x, y) -> if(x, y, if(y, false(), true()))}
Weak Trs:
{ implies(x, y) -> if(x, y, true())
, or(x, y) -> if(x, true(), y)
, and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> 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, x) -> true()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(if) = {3}, Uargs(and) = {}, Uargs(or) = {},
Uargs(implies) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
false() = [0]
[0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
or(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [2]
implies(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
=(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ not(x) -> if(x, false(), true())
, =(x, y) -> if(x, y, not(y))
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, =(x, y) -> if(x, y, if(y, false(), true()))}
Weak Trs:
{ =(x, x) -> true()
, implies(x, y) -> if(x, y, true())
, or(x, y) -> if(x, true(), y)
, and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> 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) -> if(x, y, not(y))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(if) = {3}, Uargs(and) = {}, Uargs(or) = {},
Uargs(implies) = {}, Uargs(=) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
not(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
false() = [0]
[0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
or(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [2]
implies(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
=(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ not(x) -> if(x, false(), true())
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, =(x, y) -> if(x, y, if(y, false(), true()))}
Weak Trs:
{ =(x, y) -> if(x, y, not(y))
, =(x, x) -> true()
, implies(x, y) -> if(x, y, true())
, or(x, y) -> if(x, true(), y)
, and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> 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) -> if(x, false(), true())
, =(x, y) -> if(x, y, if(y, false(), true()))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(if) = {3}, Uargs(and) = {}, Uargs(or) = {},
Uargs(implies) = {}, 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] [0]
if(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1]
[0 0] [0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [0]
or(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
implies(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [0]
=(x1, x2) = [0 0] x1 + [1 1] x2 + [3]
[0 0] [0 0] [0]
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) -> y}
Weak Trs:
{ not(x) -> if(x, false(), true())
, =(x, y) -> if(x, y, if(y, false(), true()))
, =(x, y) -> if(x, y, not(y))
, =(x, x) -> true()
, implies(x, y) -> if(x, y, true())
, or(x, y) -> if(x, true(), y)
, and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> true()}
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) -> y}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(not) = {}, Uargs(if) = {3}, Uargs(and) = {}, Uargs(or) = {},
Uargs(implies) = {}, 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]
if(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1]
[0 0] [0 0] [0 1] [1]
false() = [0]
[0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [1]
or(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [2]
implies(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
=(x1, x2) = [0 0] x1 + [1 1] x2 + [3]
[0 0] [0 0] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {if(true(), x, y) -> x}
Weak Trs:
{ if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, =(x, y) -> if(x, y, if(y, false(), true()))
, =(x, y) -> if(x, y, not(y))
, =(x, x) -> true()
, implies(x, y) -> if(x, y, true())
, or(x, y) -> if(x, true(), y)
, and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> true()}
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(not) = {}, Uargs(if) = {3}, Uargs(and) = {}, Uargs(or) = {},
Uargs(implies) = {}, 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]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [1 1] [0 1] [1]
false() = [0]
[0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [1 1] [2]
or(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [2]
implies(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [1 1] [2]
=(x1, x2) = [0 0] x1 + [1 0] x2 + [3]
[0 0] [1 1] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, =(x, y) -> if(x, y, if(y, false(), true()))
, =(x, y) -> if(x, y, not(y))
, =(x, x) -> true()
, implies(x, y) -> if(x, y, true())
, or(x, y) -> if(x, true(), y)
, and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, =(x, y) -> if(x, y, if(y, false(), true()))
, =(x, y) -> if(x, y, not(y))
, =(x, x) -> true()
, implies(x, y) -> if(x, y, true())
, or(x, y) -> if(x, true(), y)
, and(x, y) -> if(x, y, false())
, if(x, x, if(x, false(), true())) -> 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))