We consider the following Problem:
Strict Trs:
{ g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)
, h(d()) -> c(a())
, h(d()) -> c(b())
, f(k(a()), k(b()), X) -> f(X, X, X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)
, h(d()) -> c(a())
, h(d()) -> c(b())
, f(k(a()), k(b()), X) -> f(X, 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:
{ g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {1, 2}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
g(x1) = [1 1] x1 + [2]
[0 0] [2]
u(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 1] x3 + [1]
[0 0] [0 0] [0 0] [1]
h(x1) = [0 0] x1 + [0]
[0 0] [1]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[1 1] [1]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 1] x3 + [1]
[0 0] [1 1] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())
, f(k(a()), k(b()), X) -> f(X, X, X)}
Weak Trs:
{ g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {1, 2}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
g(x1) = [1 0] x1 + [3]
[1 1] [2]
u(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [0 0] [1 1] [1]
h(x1) = [0 0] x1 + [1]
[1 1] [0]
d() = [2]
[3]
c(x1) = [0 0] x1 + [0]
[0 0] [1]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [1]
[1 1] [0 0] [1 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {f(k(a()), k(b()), X) -> f(X, X, X)}
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())
, g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {f(k(a()), k(b()), X) -> f(X, X, X)}
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())
, g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs: {f^#(k(a()), k(b()), X) -> f^#(X, X, X)}
Weak DPs:
{ h^#(d()) -> c_2()
, h^#(d()) -> c_3()
, g^#(X) -> u^#(h(X), h(X), X)
, u^#(d(), c(Y), X) -> c_5()}
We consider the following Problem:
Strict DPs: {f^#(k(a()), k(b()), X) -> f^#(X, X, X)}
Strict Trs: {f(k(a()), k(b()), X) -> f(X, X, X)}
Weak DPs:
{ h^#(d()) -> c_2()
, h^#(d()) -> c_3()
, g^#(X) -> u^#(h(X), h(X), X)
, u^#(d(), c(Y), X) -> c_5()}
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())
, g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Weak Usable Rules:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
We consider the following Problem:
Strict DPs: {f^#(k(a()), k(b()), X) -> f^#(X, X, X)}
Weak DPs:
{ h^#(d()) -> c_2()
, h^#(d()) -> c_3()
, g^#(X) -> u^#(h(X), h(X), X)
, u^#(d(), c(Y), X) -> c_5()}
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {f^#(k(a()), k(b()), X) -> f^#(X, X, X)}
Weak DPs:
{ h^#(d()) -> c_2()
, h^#(d()) -> c_3()
, g^#(X) -> u^#(h(X), h(X), X)
, u^#(d(), c(Y), X) -> c_5()}
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->5:{1} [ YES(O(1),O(1)) ]
->4:{2} [ YES(O(1),O(1)) ]
->3:{3} [ YES(O(1),O(1)) ]
->2:{4} [ YES(O(1),O(1)) ]
->1:{5} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: f^#(k(a()), k(b()), X) -> f^#(X, X, X)}
WeakDPs DPs:
{ 2: h^#(d()) -> c_2()
, 3: h^#(d()) -> c_3()
, 4: g^#(X) -> u^#(h(X), h(X), X)
, 5: u^#(d(), c(Y), X) -> c_5()}
* Path 5:{1}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict DPs: {f^#(k(a()), k(b()), X) -> f^#(X, X, X)}
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: f^#(k(a()), k(b()), X) -> f^#(X, X, X)
together with the congruence-graph
->1:{1} Noncyclic, trivial, SCC
Here dependency-pairs are as follows:
Strict DPs:
{1: f^#(k(a()), k(b()), X) -> f^#(X, X, X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: f^#(k(a()), k(b()), X) -> f^#(X, X, X)}
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
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:{2}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
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:{3}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
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:{4}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
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:{5}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
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))