We consider the following Problem:
Strict Trs:
{ h(X, Z) -> f(X, s(X), Z)
, f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ h(X, Z) -> f(X, s(X), Z)
, f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {g(0(), Y) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
h(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
f(x1, x2, x3) = [1 1] x1 + [0 1] x2 + [0 0] x3 + [1]
[0 0] [1 0] [0 0] [1]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ h(X, Z) -> f(X, s(X), Z)
, f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(X, s(Y)) -> g(X, Y)}
Weak Trs: {g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(X, Y, g(X, Y)) -> h(0(), g(X, Y))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
h(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
f(x1, x2, x3) = [1 1] x1 + [0 3] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [1]
s(x1) = [0 0] x1 + [0]
[0 0] [2]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ h(X, Z) -> f(X, s(X), Z)
, g(X, s(Y)) -> g(X, Y)}
Weak Trs:
{ f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {g(X, s(Y)) -> g(X, Y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
[0 0] [0 0] [1]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 3] x3 + [1]
[0 1] [0 0] [0 0] [1]
s(x1) = [1 2] x1 + [2]
[0 0] [3]
g(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [2]
0() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {h(X, Z) -> f(X, s(X), Z)}
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {h(X, Z) -> f(X, s(X), Z)}
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs: {h^#(X, Z) -> f^#(X, s(X), Z)}
Weak DPs:
{ g^#(X, s(Y)) -> g^#(X, Y)
, f^#(X, Y, g(X, Y)) -> h^#(0(), g(X, Y))
, g^#(0(), Y) -> c_4()}
We consider the following Problem:
Strict DPs: {h^#(X, Z) -> f^#(X, s(X), Z)}
Strict Trs: {h(X, Z) -> f(X, s(X), Z)}
Weak DPs:
{ g^#(X, s(Y)) -> g^#(X, Y)
, f^#(X, Y, g(X, Y)) -> h^#(0(), g(X, Y))
, g^#(0(), Y) -> c_4()}
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Weak Usable Rules:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 0()}
We consider the following Problem:
Strict DPs: {h^#(X, Z) -> f^#(X, s(X), Z)}
Weak DPs:
{ g^#(X, s(Y)) -> g^#(X, Y)
, f^#(X, Y, g(X, Y)) -> h^#(0(), g(X, Y))
, g^#(0(), Y) -> c_4()}
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {h^#(X, Z) -> f^#(X, s(X), Z)}
Weak DPs:
{ g^#(X, s(Y)) -> g^#(X, Y)
, f^#(X, Y, g(X, Y)) -> h^#(0(), g(X, Y))
, g^#(0(), Y) -> c_4()}
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->3:{1,3} [ YES(O(1),O(1)) ]
->1:{2} [ subsumed ]
|
`->2:{4} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: h^#(X, Z) -> f^#(X, s(X), Z)}
WeakDPs DPs:
{ 2: g^#(X, s(Y)) -> g^#(X, Y)
, 3: f^#(X, Y, g(X, Y)) -> h^#(0(), g(X, Y))
, 4: g^#(0(), Y) -> c_4()}
* Path 3:{1,3}: YES(O(1),O(1))
----------------------------
We consider the following Problem:
Strict DPs: {h^#(X, Z) -> f^#(X, s(X), Z)}
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: h^#(X, Z) -> f^#(X, s(X), Z)
together with the congruence-graph
->1:{1} Noncyclic, trivial, SCC
Here dependency-pairs are as follows:
Strict DPs:
{1: h^#(X, Z) -> f^#(X, s(X), Z)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: h^#(X, Z) -> f^#(X, s(X), Z)}
We consider the following Problem:
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 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:{2}: subsumed
--------------------
This path is subsumed by the proof of paths 1:{2}->2:{4}.
* Path 1:{2}->2:{4}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {g^#(X, s(Y)) -> g^#(X, Y)}
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: g^#(X, s(Y)) -> g^#(X, Y) -->_1 g^#(X, s(Y)) -> g^#(X, Y) :1
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: g^#(X, s(Y)) -> g^#(X, Y)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: g^#(X, s(Y)) -> g^#(X, Y)}
We consider the following Problem:
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ g(X, s(Y)) -> g(X, Y)
, g(0(), Y) -> 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))