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