We consider the following Problem:
Strict Trs:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
We consider the following Problem:
Strict Trs:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(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: {-(0(), y) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(-) = {2}, Uargs(s) = {1}, Uargs(if) = {2},
Uargs(greater) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
-(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[1 0] [1 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [1]
greater(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(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:
{ -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
Weak Trs: {-(0(), y) -> 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, 0()) -> x}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(-) = {2}, Uargs(s) = {1}, Uargs(if) = {2},
Uargs(greater) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
-(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [1 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [1]
greater(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(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:
{ -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
Weak Trs:
{ -(x, 0()) -> x
, -(0(), y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {p(s(x)) -> x}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(-) = {2}, Uargs(s) = {1}, Uargs(if) = {2},
Uargs(greater) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
-(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [1 0] [0]
0() = [1]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 0] x3 + [3]
[0 0] [0 0] [1 0] [0]
greater(x1, x2) = [0 0] x1 + [0 0] x2 + [3]
[0 0] [0 0] [0]
p(x1) = [1 0] x1 + [0]
[0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()}
Weak Trs:
{ p(s(x)) -> x
, -(x, 0()) -> x
, -(0(), y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {p(0()) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(-) = {2}, Uargs(s) = {1}, Uargs(if) = {2},
Uargs(greater) = {}, Uargs(p) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
-(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [1]
0() = [2]
[3]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 2] x3 + [3]
[0 0] [0 0] [0 0] [1]
greater(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [0]
p(x1) = [1 0] x1 + [2]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())}
Weak Trs:
{ p(0()) -> 0()
, p(s(x)) -> x
, -(x, 0()) -> x
, -(0(), y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
We consider the following Problem:
Strict Trs:
{-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())}
Weak Trs:
{ p(0()) -> 0()
, p(s(x)) -> x
, -(x, 0()) -> x
, -(0(), y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The following argument positions are usable:
Uargs(-) = {2}, Uargs(s) = {1}, Uargs(if) = {2},
Uargs(greater) = {}, Uargs(p) = {}
We have the following constructor-based EDA-non-satisfying and IDA(2)-non-satisfying matrix interpretation:
Interpretation Functions:
-(x1, x2) = [1 0 0] x1 + [2 0 0] x2 + [0]
[0 2 0] [1 0 1] [0]
[0 1 1] [2 0 0] [0]
0() = [1]
[2]
[1]
s(x1) = [1 0 0] x1 + [2]
[0 1 1] [0]
[1 0 0] [0]
if(x1, x2, x3) = [0 0 0] x1 + [1 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 1] [0]
[0 0 0] [0 0 0] [1 0 0] [0]
greater(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
p(x1) = [0 0 1] x1 + [0]
[0 1 0] [2]
[0 1 0] [1]
Hurray, we answered YES(?,O(n^2))