We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, p(0()) -> 0()
, p(s(x)) -> x}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, p(0()) -> 0()
, p(s(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: {p(0()) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
cond2(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
even(x1) = [0 0] x1 + [0]
[0 0] [1]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
0() = [1]
[0]
div2(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
p(x1) = [1 0] x1 + [2]
[0 0] [3]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
y() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, p(s(x)) -> x}
Weak Trs: {p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {cond2(false(), x) -> cond1(neq(x, 0()), p(x))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
cond2(x1, x2) = [1 2] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[2]
p(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
y() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, p(s(x)) -> x}
Weak Trs:
{ cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ div2(0()) -> 0()
, div2(s(0())) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
cond2(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
even(x1) = [0 0] x1 + [0]
[0 0] [1]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [1]
[0 0] [0]
false() = [3]
[0]
p(x1) = [1 0] x1 + [1]
[0 0] [1]
s(x1) = [1 1] x1 + [0]
[0 0] [1]
y() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(s(s(x))) -> s(div2(x))
, p(s(x)) -> x}
Weak Trs:
{ div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
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(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
cond2(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
even(x1) = [0 0] x1 + [0]
[0 0] [1]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [1]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
p(x1) = [1 0] x1 + [0]
[0 1] [0]
s(x1) = [1 0] x1 + [1]
[0 1] [1]
y() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(s(s(x))) -> s(div2(x))}
Weak Trs:
{ p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
cond2(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
even(x1) = [0 0] x1 + [0]
[0 0] [1]
neq(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [2]
[0]
p(x1) = [1 0] x1 + [1]
[0 1] [0]
s(x1) = [1 0] x1 + [2]
[0 1] [1]
y() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(s(s(x))) -> s(div2(x))}
Weak Trs:
{ neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {neq(s(x), 0()) -> true()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
cond2(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
even(x1) = [0 0] x1 + [0]
[0 1] [0]
neq(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[1]
div2(x1) = [0 0] x1 + [2]
[0 0] [2]
false() = [3]
[0]
p(x1) = [1 0] x1 + [0]
[0 1] [0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
y() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, neq(0(), 0()) -> false()
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(s(s(x))) -> s(div2(x))}
Weak Trs:
{ neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {neq(0(), 0()) -> false()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 2] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
cond2(x1, x2) = [1 3] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [2]
0() = [0]
[2]
div2(x1) = [0 0] x1 + [1]
[0 0] [3]
false() = [2]
[2]
p(x1) = [1 0] x1 + [0]
[0 1] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
y() = [0]
[2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(s(s(x))) -> s(div2(x))}
Weak Trs:
{ neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ cond1(true(), x) -> cond2(even(x), x)
, div2(s(s(x))) -> s(div2(x))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[3]
cond2(x1, x2) = [1 3] x1 + [1 1] x2 + [3]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[0 0] [0 1] [0]
0() = [0]
[3]
div2(x1) = [1 0] x1 + [2]
[0 1] [2]
false() = [2]
[2]
p(x1) = [1 0] x1 + [0]
[0 1] [0]
s(x1) = [1 0] x1 + [2]
[0 1] [3]
y() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ cond1(true(), x) -> cond2(even(x), x)
, div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {cond2(true(), x) -> cond1(neq(x, 0()), div2(x))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(cond1) = {1, 2}, Uargs(cond2) = {1}, Uargs(even) = {},
Uargs(neq) = {}, Uargs(div2) = {}, Uargs(p) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
true() = [1]
[0]
cond2(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [1]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
0() = [1]
[0]
div2(x1) = [1 0] x1 + [0]
[0 0] [1]
false() = [0]
[0]
p(x1) = [1 0] x1 + [0]
[0 1] [1]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
y() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, cond1(true(), x) -> cond2(even(x), x)
, div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, cond1(true(), x) -> cond2(even(x), x)
, div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs:
{ even^#(0()) -> c_1()
, even^#(s(0())) -> c_2()
, even^#(s(s(x))) -> even^#(x)}
Weak DPs:
{ cond2^#(true(), x) -> cond1^#(neq(x, 0()), div2(x))
, cond1^#(true(), x) -> cond2^#(even(x), x)
, div2^#(s(s(x))) -> div2^#(x)
, neq^#(0(), 0()) -> c_7()
, neq^#(s(x), 0()) -> c_8()
, neq^#(0(), s(x)) -> c_9()
, neq^#(s(x), s(y())) -> neq^#(x, y())
, p^#(s(x)) -> c_11()
, div2^#(0()) -> c_12()
, div2^#(s(0())) -> c_13()
, cond2^#(false(), x) -> cond1^#(neq(x, 0()), p(x))
, p^#(0()) -> c_15()}
We consider the following Problem:
Strict DPs:
{ even^#(0()) -> c_1()
, even^#(s(0())) -> c_2()
, even^#(s(s(x))) -> even^#(x)}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs:
{ cond2^#(true(), x) -> cond1^#(neq(x, 0()), div2(x))
, cond1^#(true(), x) -> cond2^#(even(x), x)
, div2^#(s(s(x))) -> div2^#(x)
, neq^#(0(), 0()) -> c_7()
, neq^#(s(x), 0()) -> c_8()
, neq^#(0(), s(x)) -> c_9()
, neq^#(s(x), s(y())) -> neq^#(x, y())
, p^#(s(x)) -> c_11()
, div2^#(0()) -> c_12()
, div2^#(s(0())) -> c_13()
, cond2^#(false(), x) -> cond1^#(neq(x, 0()), p(x))
, p^#(0()) -> c_15()}
Weak Trs:
{ cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, cond1(true(), x) -> cond2(even(x), x)
, div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Usable Rules:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
We consider the following Problem:
Strict DPs:
{ even^#(0()) -> c_1()
, even^#(s(0())) -> c_2()
, even^#(s(s(x))) -> even^#(x)}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs:
{ cond2^#(true(), x) -> cond1^#(neq(x, 0()), div2(x))
, cond1^#(true(), x) -> cond2^#(even(x), x)
, div2^#(s(s(x))) -> div2^#(x)
, neq^#(0(), 0()) -> c_7()
, neq^#(s(x), 0()) -> c_8()
, neq^#(0(), s(x)) -> c_9()
, neq^#(s(x), s(y())) -> neq^#(x, y())
, p^#(s(x)) -> c_11()
, div2^#(0()) -> c_12()
, div2^#(s(0())) -> c_13()
, cond2^#(false(), x) -> cond1^#(neq(x, 0()), p(x))
, p^#(0()) -> c_15()}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs:
{ even^#(0()) -> c_1()
, even^#(s(0())) -> c_2()
, even^#(s(s(x))) -> even^#(x)}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs:
{ cond2^#(true(), x) -> cond1^#(neq(x, 0()), div2(x))
, cond1^#(true(), x) -> cond2^#(even(x), x)
, div2^#(s(s(x))) -> div2^#(x)
, neq^#(0(), 0()) -> c_7()
, neq^#(s(x), 0()) -> c_8()
, neq^#(0(), s(x)) -> c_9()
, neq^#(s(x), s(y())) -> neq^#(x, y())
, p^#(s(x)) -> c_11()
, div2^#(0()) -> c_12()
, div2^#(s(0())) -> c_13()
, cond2^#(false(), x) -> cond1^#(neq(x, 0()), p(x))
, p^#(0()) -> c_15()}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->11:{3} [ YES(?,O(n^1)) ]
|
|->13:{1} [ YES(?,O(n^1)) ]
|
`->12:{2} [ YES(?,O(n^1)) ]
->10:{4,5,14} [ YES(O(1),O(1)) ]
->7:{6} [ subsumed ]
|
|->8:{12} [ YES(O(1),O(1)) ]
|
`->9:{13} [ YES(O(1),O(1)) ]
->6:{7} [ YES(O(1),O(1)) ]
->5:{8} [ YES(O(1),O(1)) ]
->4:{9} [ YES(O(1),O(1)) ]
->3:{10} [ YES(O(1),O(1)) ]
->2:{11} [ YES(O(1),O(1)) ]
->1:{15} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{ 1: even^#(0()) -> c_1()
, 2: even^#(s(0())) -> c_2()
, 3: even^#(s(s(x))) -> even^#(x)}
WeakDPs DPs:
{ 4: cond2^#(true(), x) -> cond1^#(neq(x, 0()), div2(x))
, 5: cond1^#(true(), x) -> cond2^#(even(x), x)
, 6: div2^#(s(s(x))) -> div2^#(x)
, 7: neq^#(0(), 0()) -> c_7()
, 8: neq^#(s(x), 0()) -> c_8()
, 9: neq^#(0(), s(x)) -> c_9()
, 10: neq^#(s(x), s(y())) -> neq^#(x, y())
, 11: p^#(s(x)) -> c_11()
, 12: div2^#(0()) -> c_12()
, 13: div2^#(s(0())) -> c_13()
, 14: cond2^#(false(), x) -> cond1^#(neq(x, 0()), p(x))
, 15: p^#(0()) -> c_15()}
* Path 11:{3}: YES(?,O(n^1))
--------------------------
We consider the following Problem:
Strict DPs: {even^#(s(s(x))) -> even^#(x)}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {even^#(s(s(x))) -> even^#(x)}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {even^#(s(s(x))) -> even^#(x)}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {even^#(s(s(x))) -> even^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ s_0(2) -> 2
, even^#_0(2) -> 1
, even^#_1(2) -> 1}
* Path 11:{3}->13:{1}: YES(?,O(n^1))
----------------------------------
We consider the following Problem:
Strict DPs: {even^#(0()) -> c_1()}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {even^#(s(s(x))) -> even^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {even^#(0()) -> c_1()}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {even^#(s(s(x))) -> even^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {even^#(0()) -> c_1()}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {even^#(s(s(x))) -> even^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {even^#(0()) -> c_1()}
Weak DPs: {even^#(s(s(x))) -> even^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 0_0() -> 2
, s_0(2) -> 2
, even^#_0(2) -> 1
, c_1_1() -> 1}
* Path 11:{3}->12:{2}: YES(?,O(n^1))
----------------------------------
We consider the following Problem:
Strict DPs: {even^#(s(0())) -> c_2()}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {even^#(s(s(x))) -> even^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {even^#(s(0())) -> c_2()}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {even^#(s(s(x))) -> even^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {even^#(s(0())) -> c_2()}
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {even^#(s(s(x))) -> even^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {even^#(s(0())) -> c_2()}
Weak DPs: {even^#(s(s(x))) -> even^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 0_0() -> 2
, s_0(2) -> 2
, even^#_0(2) -> 1
, c_2_1() -> 1}
* Path 10:{4,5,14}: YES(O(1),O(1))
--------------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 7:{6}: subsumed
--------------------
This path is subsumed by the proof of paths 7:{6}->9:{13},
7:{6}->8:{12}.
* Path 7:{6}->8:{12}: YES(O(1),O(1))
----------------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {div2^#(s(s(x))) -> div2^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {div2^#(s(s(x))) -> div2^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {div2^#(s(s(x))) -> div2^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {div2^#(s(s(x))) -> div2^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 7:{6}->9:{13}: YES(O(1),O(1))
----------------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {div2^#(s(s(x))) -> div2^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {div2^#(s(s(x))) -> div2^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak DPs: {div2^#(s(s(x))) -> div2^#(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {div2^#(s(s(x))) -> div2^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 6:{7}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{8}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 4:{9}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 3:{10}: YES(O(1),O(1))
---------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 2:{11}: YES(O(1),O(1))
---------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{15}: YES(O(1),O(1))
---------------------------
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)}
Weak Trs:
{ div2(s(s(x))) -> s(div2(x))
, neq(0(), 0()) -> false()
, neq(s(x), 0()) -> true()
, neq(0(), s(x)) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, p(s(x)) -> x
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, p(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))