We consider the following Problem:
Strict Trs:
{ f(true(), x, y, z) -> g(gt(x, y), x, y, z)
, g(true(), x, y, z) -> f(gt(x, z), x, s(y), z)
, g(true(), x, y, z) -> f(gt(x, z), x, y, s(z))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(true(), x, y, z) -> g(gt(x, y), x, y, z)
, g(true(), x, y, z) -> f(gt(x, z), x, s(y), z)
, g(true(), x, y, z) -> f(gt(x, z), x, y, s(z))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ g(true(), x, y, z) -> f(gt(x, z), x, s(y), z)
, g(true(), x, y, z) -> f(gt(x, z), x, y, s(z))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(g) = {1}, Uargs(gt) = {}, Uargs(s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1 1] x4 + [1]
[0 0] [0 0] [0 0] [0 0] [1]
true() = [0]
[2]
g(x1, x2, x3, x4) = [1 1] x1 + [1 1] x2 + [1 1] x3 + [1 1] x4 + [1]
[0 0] [0 0] [0 0] [0 0] [1]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(true(), x, y, z) -> g(gt(x, y), x, y, z)
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs:
{ g(true(), x, y, z) -> f(gt(x, z), x, s(y), z)
, g(true(), x, y, z) -> f(gt(x, z), x, y, s(z))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(true(), x, y, z) -> g(gt(x, y), x, y, z)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(g) = {1}, Uargs(gt) = {}, Uargs(s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1 1] x4 + [1]
[0 0] [0 0] [0 0] [0 0] [1]
true() = [0]
[3]
g(x1, x2, x3, x4) = [1 3] x1 + [1 1] x2 + [1 1] x3 + [1 1] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [1]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs:
{ f(true(), x, y, z) -> g(gt(x, y), x, y, z)
, g(true(), x, y, z) -> f(gt(x, z), x, s(y), z)
, g(true(), x, y, z) -> f(gt(x, z), x, y, s(z))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {gt(0(), v) -> false()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(g) = {1}, Uargs(gt) = {}, Uargs(s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3, x4) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1 1] x4 + [3]
[0 0] [0 0] [0 0] [0 0] [1]
true() = [0]
[3]
g(x1, x2, x3, x4) = [1 1] x1 + [1 1] x2 + [1 1] x3 + [1 1] x4 + [1]
[0 0] [0 0] [0 0] [0 0] [1]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
s(x1) = [0 1] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs:
{ gt(0(), v) -> false()
, f(true(), x, y, z) -> g(gt(x, y), x, y, z)
, g(true(), x, y, z) -> f(gt(x, z), x, s(y), z)
, g(true(), x, y, z) -> f(gt(x, z), x, y, s(z))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs:
{ gt(0(), v) -> false()
, f(true(), x, y, z) -> g(gt(x, y), x, y, z)
, g(true(), x, y, z) -> f(gt(x, z), x, s(y), z)
, g(true(), x, y, z) -> f(gt(x, z), x, y, s(z))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs:
{ gt^#(s(u), 0()) -> c_1()
, gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak DPs:
{ gt^#(0(), v) -> c_3()
, f^#(true(), x, y, z) -> g^#(gt(x, y), x, y, z)
, g^#(true(), x, y, z) -> f^#(gt(x, z), x, s(y), z)
, g^#(true(), x, y, z) -> f^#(gt(x, z), x, y, s(z))}
We consider the following Problem:
Strict DPs:
{ gt^#(s(u), 0()) -> c_1()
, gt^#(s(u), s(v)) -> gt^#(u, v)}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs:
{ gt^#(0(), v) -> c_3()
, f^#(true(), x, y, z) -> g^#(gt(x, y), x, y, z)
, g^#(true(), x, y, z) -> f^#(gt(x, z), x, s(y), z)
, g^#(true(), x, y, z) -> f^#(gt(x, z), x, y, s(z))}
Weak Trs:
{ gt(0(), v) -> false()
, f(true(), x, y, z) -> g(gt(x, y), x, y, z)
, g(true(), x, y, z) -> f(gt(x, z), x, s(y), z)
, g(true(), x, y, z) -> f(gt(x, z), x, y, s(z))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Usable Rules: {gt(0(), v) -> false()}
We consider the following Problem:
Strict DPs:
{ gt^#(s(u), 0()) -> c_1()
, gt^#(s(u), s(v)) -> gt^#(u, v)}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs:
{ gt^#(0(), v) -> c_3()
, f^#(true(), x, y, z) -> g^#(gt(x, y), x, y, z)
, g^#(true(), x, y, z) -> f^#(gt(x, z), x, s(y), z)
, g^#(true(), x, y, z) -> f^#(gt(x, z), x, y, s(z))}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs:
{ gt^#(s(u), 0()) -> c_1()
, gt^#(s(u), s(v)) -> gt^#(u, v)}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs:
{ gt^#(0(), v) -> c_3()
, f^#(true(), x, y, z) -> g^#(gt(x, y), x, y, z)
, g^#(true(), x, y, z) -> f^#(gt(x, z), x, s(y), z)
, g^#(true(), x, y, z) -> f^#(gt(x, z), x, y, s(z))}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->2:{2} [ YES(?,O(n^1)) ]
|
|->4:{1} [ YES(?,O(n^1)) ]
|
`->3:{3} [ YES(O(1),O(1)) ]
->1:{4,6,5} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{ 1: gt^#(s(u), 0()) -> c_1()
, 2: gt^#(s(u), s(v)) -> gt^#(u, v)}
WeakDPs DPs:
{ 3: gt^#(0(), v) -> c_3()
, 4: f^#(true(), x, y, z) -> g^#(gt(x, y), x, y, z)
, 5: g^#(true(), x, y, z) -> f^#(gt(x, z), x, s(y), z)
, 6: g^#(true(), x, y, z) -> f^#(gt(x, z), x, y, s(z))}
* Path 2:{2}: YES(?,O(n^1))
-------------------------
We consider the following Problem:
Strict DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
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
, gt^#_0(2, 2) -> 1
, gt^#_1(2, 2) -> 1}
* Path 2:{2}->4:{1}: YES(?,O(n^1))
--------------------------------
We consider the following Problem:
Strict DPs: {gt^#(s(u), 0()) -> c_1()}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {gt^#(s(u), 0()) -> c_1()}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {gt^#(s(u), 0()) -> c_1()}
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {gt^#(s(u), 0()) -> c_1()}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
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
, 0_0() -> 2
, gt^#_0(2, 2) -> 1
, c_1_1() -> 1}
* Path 2:{2}->3:{3}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{4,6,5}: YES(O(1),O(1))
------------------------------
We consider the following Problem:
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs: {gt(0(), v) -> false()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs: {gt(0(), v) -> false()}
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))