* Step 1: Sum WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
g(a(),g(x,g(b(),g(a(),g(x,y))))) -> g(a(),g(a(),g(a(),g(x,g(b(),g(b(),y))))))
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2} / {a/0,b/0,c/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g,h,i} and constructors {a,b,c,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
g(a(),g(x,g(b(),g(a(),g(x,y))))) -> g(a(),g(a(),g(a(),g(x,g(b(),g(b(),y))))))
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2} / {a/0,b/0,c/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g,h,i} and constructors {a,b,c,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
g#(x,x) -> c_3(g#(a(),b()))
g#(a(),g(x,g(b(),g(a(),g(x,y))))) -> c_4(g#(a(),g(a(),g(a(),g(x,g(b(),g(b(),y))))))
,g#(a(),g(a(),g(x,g(b(),g(b(),y)))))
,g#(a(),g(x,g(b(),g(b(),y))))
,g#(x,g(b(),g(b(),y)))
,g#(b(),g(b(),y))
,g#(b(),y))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
i#(x,x) -> c_8(i#(a(),b()))
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
g#(x,x) -> c_3(g#(a(),b()))
g#(a(),g(x,g(b(),g(a(),g(x,y))))) -> c_4(g#(a(),g(a(),g(a(),g(x,g(b(),g(b(),y))))))
,g#(a(),g(a(),g(x,g(b(),g(b(),y)))))
,g#(a(),g(x,g(b(),g(b(),y))))
,g#(x,g(b(),g(b(),y)))
,g#(b(),g(b(),y))
,g#(b(),y))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
i#(x,x) -> c_8(i#(a(),b()))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
g(a(),g(x,g(b(),g(a(),g(x,y))))) -> g(a(),g(a(),g(a(),g(x,g(b(),g(b(),y))))))
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/2,c_2/2,c_3/1,c_4/6,c_5/2,c_6/4,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
g#(x,x) -> c_3(g#(a(),b()))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
i#(x,x) -> c_8(i#(a(),b()))
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
g#(x,x) -> c_3(g#(a(),b()))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
i#(x,x) -> c_8(i#(a(),b()))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/2,c_2/2,c_3/1,c_4/6,c_5/2,c_6/4,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{3,7}
by application of
Pre({3,7}) = {1,4,5}.
Here rules are labelled as follows:
1: f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
2: f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
3: g#(x,x) -> c_3(g#(a(),b()))
4: h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
5: h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
6: h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
7: i#(x,x) -> c_8(i#(a(),b()))
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
g#(x,x) -> c_3(g#(a(),b()))
i#(x,x) -> c_8(i#(a(),b()))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/2,c_2/2,c_3/1,c_4/6,c_5/2,c_6/4,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
-->_1 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_2 g#(x,x) -> c_3(g#(a(),b())):6
-->_1 f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y))):1
2:S:f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
-->_2 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_1 f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y))):1
3:S:h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
-->_1 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y)):4
-->_2 g#(x,x) -> c_3(g#(a(),b())):6
-->_1 h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y)):3
4:S:h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
-->_4 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_3 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_2 i#(x,x) -> c_8(i#(a(),b())):7
-->_1 i#(x,x) -> c_8(i#(a(),b())):7
-->_4 h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y)):4
-->_3 h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y)):4
-->_4 h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y)):3
-->_3 h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y)):3
5:S:h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
-->_1 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y)):4
-->_1 h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y)):3
-->_2 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_2 f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y))):1
6:W:g#(x,x) -> c_3(g#(a(),b()))
7:W:i#(x,x) -> c_8(i#(a(),b()))
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: g#(x,x) -> c_3(g#(a(),b()))
7: i#(x,x) -> c_8(i#(a(),b()))
* Step 6: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/2,c_2/2,c_3/1,c_4/6,c_5/2,c_6/4,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y)))
-->_1 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_1 f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y))):1
2:S:f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
-->_2 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_1 f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y))):1
3:S:h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y))
-->_1 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y)):4
-->_1 h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y)):3
4:S:h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y))
-->_4 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_3 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_4 h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y)):4
-->_3 h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y)):4
-->_4 h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y)):3
-->_3 h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y)):3
5:S:h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
-->_1 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 h#(i(x,y)) -> c_6(i#(i(c(),h(h(y))),x),i#(c(),h(h(y))),h#(h(y)),h#(y)):4
-->_1 h#(g(x,s(y))) -> c_5(h#(g(s(x),y)),g#(s(x),y)):3
-->_2 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_2 f#(g(s(x),y)) -> c_1(f#(g(x,s(y))),g#(x,s(y))):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
* Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
The strictly oriented rules are moved into the weak component.
** Step 7.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_2) = {1,2},
uargs(c_5) = {1},
uargs(c_6) = {1,2},
uargs(c_7) = {2}
Following symbols are considered usable:
{f,g,h,i,f#,g#,h#,i#}
TcT has computed the following interpretation:
p(a) = [0]
[0]
[0]
p(b) = [0]
[0]
[0]
p(c) = [0]
[0]
[0]
p(f) = [1 0 0] [1]
[1 1 0] x1 + [1]
[1 1 1] [0]
p(g) = [0 0 0] [0]
[0 0 0] x2 + [0]
[0 0 1] [0]
p(h) = [0 0 0] [0]
[0 0 0] x1 + [0]
[1 1 0] [0]
p(i) = [0 0 1] [0 0 0] [0]
[1 1 1] x1 + [0 0 0] x2 + [0]
[0 0 0] [1 1 1] [0]
p(s) = [0 1 1] [1]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(f#) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(g#) = [0]
[0]
[0]
p(h#) = [0 0 1] [0]
[0 0 1] x1 + [0]
[1 0 0] [1]
p(i#) = [0]
[0]
[0]
p(c_1) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(c_2) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
p(c_3) = [0]
[0]
[0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[1 0 0] x1 + [0]
[0 0 0] [1]
p(c_6) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
p(c_7) = [1 1 0] [0]
[1 1 0] x2 + [0]
[0 0 0] [0]
p(c_8) = [0]
[0]
[0]
Following rules are strictly oriented:
f#(s(x)) = [0 1 1] [1]
[0 0 0] x + [0]
[0 0 0] [1]
> [0 0 1] [0]
[0 0 0] x + [0]
[0 0 0] [1]
= c_2(f#(h(s(x))),h#(s(x)))
Following rules are (at-least) weakly oriented:
f#(g(s(x),y)) = [0]
[0]
[1]
>= [0]
[0]
[1]
= c_1(f#(g(x,s(y))))
h#(g(x,s(y))) = [0 0 1] [0]
[0 0 1] y + [0]
[0 0 0] [1]
>= [0 0 1] [0]
[0 0 1] y + [0]
[0 0 0] [1]
= c_5(h#(g(s(x),y)))
h#(i(x,y)) = [0 0 0] [1 1 1] [0]
[0 0 0] x + [1 1 1] y + [0]
[0 0 1] [0 0 0] [1]
>= [1 1 1] [0]
[0 0 1] y + [0]
[0 0 0] [1]
= c_6(h#(h(y)),h#(y))
h#(s(f(x))) = [1 1 1] [0]
[1 1 1] x + [0]
[2 2 1] [3]
>= [1 1 0] [0]
[1 1 0] x + [0]
[0 0 0] [0]
= c_7(h#(f(x)),f#(x))
f(g(s(x),y)) = [0 0 0] [1]
[0 0 0] y + [1]
[0 0 1] [0]
>= [0 0 0] [1]
[0 0 0] y + [1]
[0 0 1] [0]
= f(g(x,s(y)))
f(s(x)) = [0 1 1] [2]
[0 1 1] x + [2]
[0 1 2] [1]
>= [0 1 1] [2]
[0 0 0] x + [0]
[0 1 1] [1]
= s(s(f(h(s(x)))))
g(x,x) = [0 0 0] [0]
[0 0 0] x + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= g(a(),b())
h(g(x,s(y))) = [0]
[0]
[0]
>= [0]
[0]
[0]
= h(g(s(x),y))
h(i(x,y)) = [0 0 0] [0]
[0 0 0] x + [0]
[1 1 2] [0]
>= [0 0 0] [0]
[0 0 0] x + [0]
[1 1 1] [0]
= i(i(c(),h(h(y))),x)
h(s(f(x))) = [0 0 0] [0]
[0 0 0] x + [0]
[2 2 1] [2]
>= [0 0 0] [0]
[0 0 0] x + [0]
[2 1 0] [2]
= h(f(x))
i(x,x) = [0 0 1] [0]
[1 1 1] x + [0]
[1 1 1] [0]
>= [0]
[0]
[0]
= i(a(),b())
** Step 7.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 7.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
The strictly oriented rules are moved into the weak component.
*** Step 7.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_2) = {1,2},
uargs(c_5) = {1},
uargs(c_6) = {1,2},
uargs(c_7) = {2}
Following symbols are considered usable:
{f,g,h,i,f#,g#,h#,i#}
TcT has computed the following interpretation:
p(a) = [0]
[0]
p(b) = [0]
[0]
p(c) = [0]
[0]
p(f) = [0 0] x1 + [0]
[2 2] [2]
p(g) = [0 0] x1 + [0 1] x2 + [0]
[0 2] [0 0] [0]
p(h) = [0 0] x1 + [0]
[1 0] [0]
p(i) = [3 1] x1 + [0 0] x2 + [0]
[0 0] [3 1] [0]
p(s) = [0 0] x1 + [0]
[0 1] [1]
p(f#) = [2 2] x1 + [3]
[0 0] [1]
p(g#) = [2 1] x1 + [1 2] x2 + [0]
[0 0] [0 0] [1]
p(h#) = [2 1] x1 + [0]
[0 0] [1]
p(i#) = [0 0] x2 + [0]
[0 1] [1]
p(c_1) = [1 1] x1 + [0]
[0 1] [0]
p(c_2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
p(c_3) = [0]
[0]
p(c_4) = [0 0] x2 + [0 0] x3 + [0 0] x6 + [0]
[2 0] [0 2] [2 0] [1]
p(c_5) = [1 0] x1 + [0]
[0 1] [0]
p(c_6) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
p(c_7) = [1 0] x2 + [0]
[0 1] [0]
p(c_8) = [2]
[0]
Following rules are strictly oriented:
f#(g(s(x),y)) = [0 4] x + [0 2] y + [7]
[0 0] [0 0] [1]
> [0 4] x + [0 2] y + [6]
[0 0] [0 0] [1]
= c_1(f#(g(x,s(y))))
Following rules are (at-least) weakly oriented:
f#(s(x)) = [0 2] x + [5]
[0 0] [1]
>= [0 1] x + [5]
[0 0] [1]
= c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) = [0 2] x + [0 2] y + [2]
[0 0] [0 0] [1]
>= [0 2] x + [0 2] y + [2]
[0 0] [0 0] [1]
= c_5(h#(g(s(x),y)))
h#(i(x,y)) = [6 2] x + [3 1] y + [0]
[0 0] [0 0] [1]
>= [3 1] y + [0]
[0 0] [0]
= c_6(h#(h(y)),h#(y))
h#(s(f(x))) = [2 2] x + [3]
[0 0] [1]
>= [2 2] x + [3]
[0 0] [1]
= c_7(h#(f(x)),f#(x))
f(g(s(x),y)) = [0 0] x + [0 0] y + [0]
[0 4] [0 2] [6]
>= [0 0] x + [0 0] y + [0]
[0 4] [0 2] [4]
= f(g(x,s(y)))
f(s(x)) = [0 0] x + [0]
[0 2] [4]
>= [0]
[4]
= s(s(f(h(s(x)))))
g(x,x) = [0 1] x + [0]
[0 2] [0]
>= [0]
[0]
= g(a(),b())
h(g(x,s(y))) = [0 0] y + [0]
[0 1] [1]
>= [0 0] y + [0]
[0 1] [0]
= h(g(s(x),y))
h(i(x,y)) = [0 0] x + [0]
[3 1] [0]
>= [0 0] x + [0]
[3 1] [0]
= i(i(c(),h(h(y))),x)
h(s(f(x))) = [0]
[0]
>= [0]
[0]
= h(f(x))
i(x,x) = [3 1] x + [0]
[3 1] [0]
>= [0]
[0]
= i(a(),b())
*** Step 7.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 7.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
The strictly oriented rules are moved into the weak component.
**** Step 7.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_2) = {1,2},
uargs(c_5) = {1},
uargs(c_6) = {1,2},
uargs(c_7) = {2}
Following symbols are considered usable:
{f,g,h,i,f#,g#,h#,i#}
TcT has computed the following interpretation:
p(a) = [0]
[0]
p(b) = [0]
[0]
p(c) = [0]
[0]
p(f) = [3 0] x1 + [1]
[0 0] [0]
p(g) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [2 0] [1]
p(h) = [0 0] x1 + [0]
[2 0] [0]
p(i) = [1 1] x1 + [0 0] x2 + [0]
[0 0] [2 1] [0]
p(s) = [1 0] x1 + [2]
[0 0] [0]
p(f#) = [0 0] x1 + [0]
[1 1] [0]
p(g#) = [0 0] x2 + [2]
[0 1] [1]
p(h#) = [0 1] x1 + [0]
[0 0] [0]
p(i#) = [0]
[0]
p(c_1) = [2 0] x1 + [0]
[0 0] [3]
p(c_2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
p(c_3) = [2]
[0]
p(c_4) = [0 1] x2 + [0]
[2 0] [0]
p(c_5) = [1 0] x1 + [2]
[0 0] [0]
p(c_6) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
p(c_7) = [1 0] x2 + [0]
[0 0] [0]
p(c_8) = [0 0] x1 + [1]
[0 1] [1]
Following rules are strictly oriented:
h#(g(x,s(y))) = [2 0] y + [5]
[0 0] [0]
> [2 0] y + [3]
[0 0] [0]
= c_5(h#(g(s(x),y)))
Following rules are (at-least) weakly oriented:
f#(g(s(x),y)) = [0 0] x + [0 0] y + [0]
[1 0] [3 0] [3]
>= [0]
[3]
= c_1(f#(g(x,s(y))))
f#(s(x)) = [0 0] x + [0]
[1 0] [2]
>= [0]
[0]
= c_2(f#(h(s(x))),h#(s(x)))
h#(i(x,y)) = [2 1] y + [0]
[0 0] [0]
>= [2 1] y + [0]
[0 0] [0]
= c_6(h#(h(y)),h#(y))
h#(s(f(x))) = [0]
[0]
>= [0]
[0]
= c_7(h#(f(x)),f#(x))
f(g(s(x),y)) = [3 0] x + [3 0] y + [7]
[0 0] [0 0] [0]
>= [3 0] x + [3 0] y + [7]
[0 0] [0 0] [0]
= f(g(x,s(y)))
f(s(x)) = [3 0] x + [7]
[0 0] [0]
>= [5]
[0]
= s(s(f(h(s(x)))))
g(x,x) = [2 0] x + [0]
[2 0] [1]
>= [0]
[1]
= g(a(),b())
h(g(x,s(y))) = [0 0] x + [0 0] y + [0]
[2 0] [2 0] [4]
>= [0 0] x + [0 0] y + [0]
[2 0] [2 0] [4]
= h(g(s(x),y))
h(i(x,y)) = [0 0] x + [0]
[2 2] [0]
>= [0 0] x + [0]
[2 1] [0]
= i(i(c(),h(h(y))),x)
h(s(f(x))) = [0 0] x + [0]
[6 0] [6]
>= [0 0] x + [0]
[6 0] [2]
= h(f(x))
i(x,x) = [1 1] x + [0]
[2 1] [0]
>= [0]
[0]
= i(a(),b())
**** Step 7.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 7.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
The strictly oriented rules are moved into the weak component.
***** Step 7.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_2) = {1,2},
uargs(c_5) = {1},
uargs(c_6) = {1,2},
uargs(c_7) = {2}
Following symbols are considered usable:
{f,g,h,i,f#,g#,h#,i#}
TcT has computed the following interpretation:
p(a) = [0]
[0]
p(b) = [0]
[0]
p(c) = [0]
[0]
p(f) = [3 2] x1 + [3]
[3 2] [3]
p(g) = [0 2] x1 + [0]
[0 0] [2]
p(h) = [0 0] x1 + [0]
[1 0] [0]
p(i) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [1 2] [0]
p(s) = [0 1] x1 + [1]
[0 1] [0]
p(f#) = [3 2] x1 + [2]
[0 0] [1]
p(g#) = [2 0] x2 + [0]
[0 0] [0]
p(h#) = [0 2] x1 + [0]
[0 0] [0]
p(i#) = [0 0] x2 + [0]
[1 0] [0]
p(c_1) = [1 0] x1 + [0]
[0 1] [0]
p(c_2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
p(c_3) = [1]
[0]
p(c_4) = [0 2] x1 + [0 2] x2 + [2 1] x4 + [1]
[0 2] [2 0] [0 0] [1]
p(c_5) = [1 1] x1 + [0]
[0 0] [0]
p(c_6) = [1 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
p(c_7) = [2 0] x2 + [0]
[0 0] [0]
p(c_8) = [0]
[2]
Following rules are strictly oriented:
h#(s(f(x))) = [6 4] x + [6]
[0 0] [0]
> [6 4] x + [4]
[0 0] [0]
= c_7(h#(f(x)),f#(x))
Following rules are (at-least) weakly oriented:
f#(g(s(x),y)) = [0 6] x + [6]
[0 0] [1]
>= [0 6] x + [6]
[0 0] [1]
= c_1(f#(g(x,s(y))))
f#(s(x)) = [0 5] x + [5]
[0 0] [1]
>= [0 4] x + [5]
[0 0] [1]
= c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) = [4]
[0]
>= [4]
[0]
= c_5(h#(g(s(x),y)))
h#(i(x,y)) = [2 4] y + [0]
[0 0] [0]
>= [2 4] y + [0]
[0 0] [0]
= c_6(h#(h(y)),h#(y))
f(g(s(x),y)) = [0 6] x + [7]
[0 6] [7]
>= [0 6] x + [7]
[0 6] [7]
= f(g(x,s(y)))
f(s(x)) = [0 5] x + [6]
[0 5] [6]
>= [0 2] x + [6]
[0 2] [5]
= s(s(f(h(s(x)))))
g(x,x) = [0 2] x + [0]
[0 0] [2]
>= [0]
[2]
= g(a(),b())
h(g(x,s(y))) = [0 0] x + [0]
[0 2] [0]
>= [0 0] x + [0]
[0 2] [0]
= h(g(s(x),y))
h(i(x,y)) = [0 0] x + [0]
[1 2] [0]
>= [0 0] x + [0]
[1 2] [0]
= i(i(c(),h(h(y))),x)
h(s(f(x))) = [0 0] x + [0]
[3 2] [4]
>= [0 0] x + [0]
[3 2] [3]
= h(f(x))
i(x,x) = [1 2] x + [0]
[1 2] [0]
>= [0]
[0]
= i(a(),b())
***** Step 7.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 7.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
The strictly oriented rules are moved into the weak component.
****** Step 7.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_2) = {1,2},
uargs(c_5) = {1},
uargs(c_6) = {1,2},
uargs(c_7) = {2}
Following symbols are considered usable:
{f,g,h,i,f#,g#,h#,i#}
TcT has computed the following interpretation:
p(a) = [0]
[0]
p(b) = [0]
[0]
p(c) = [0]
[0]
p(f) = [0]
[0]
p(g) = [1 2] x1 + [1]
[0 2] [0]
p(h) = [0 3] x1 + [0]
[1 0] [0]
p(i) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [1 1] [1]
p(s) = [0]
[0]
p(f#) = [0 0] x1 + [0]
[0 1] [2]
p(g#) = [0 0] x1 + [0]
[1 0] [1]
p(h#) = [0 2] x1 + [0]
[0 0] [3]
p(i#) = [0]
[0]
p(c_1) = [2 0] x1 + [0]
[0 0] [2]
p(c_2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [1]
p(c_3) = [0]
[0]
p(c_4) = [0]
[0]
p(c_5) = [1 0] x1 + [0]
[0 0] [0]
p(c_6) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
p(c_7) = [2 0] x2 + [0]
[0 0] [0]
p(c_8) = [0]
[0]
Following rules are strictly oriented:
h#(i(x,y)) = [2 2] y + [2]
[0 0] [3]
> [2 2] y + [0]
[0 0] [3]
= c_6(h#(h(y)),h#(y))
Following rules are (at-least) weakly oriented:
f#(g(s(x),y)) = [0]
[2]
>= [0]
[2]
= c_1(f#(g(x,s(y))))
f#(s(x)) = [0]
[2]
>= [0]
[1]
= c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) = [0 4] x + [0]
[0 0] [3]
>= [0]
[0]
= c_5(h#(g(s(x),y)))
h#(s(f(x))) = [0]
[3]
>= [0]
[0]
= c_7(h#(f(x)),f#(x))
f(g(s(x),y)) = [0]
[0]
>= [0]
[0]
= f(g(x,s(y)))
f(s(x)) = [0]
[0]
>= [0]
[0]
= s(s(f(h(s(x)))))
g(x,x) = [1 2] x + [1]
[0 2] [0]
>= [1]
[0]
= g(a(),b())
h(g(x,s(y))) = [0 6] x + [0]
[1 2] [1]
>= [0]
[1]
= h(g(s(x),y))
h(i(x,y)) = [0 0] x + [3 3] y + [3]
[1 1] [0 0] [1]
>= [0 0] x + [3 3] y + [3]
[1 1] [0 0] [1]
= i(i(c(),h(h(y))),x)
h(s(f(x))) = [0]
[0]
>= [0]
[0]
= h(f(x))
i(x,x) = [1 1] x + [1]
[1 1] [1]
>= [1]
[1]
= i(a(),b())
****** Step 7.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
****** Step 7.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
-->_1 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_1 f#(g(s(x),y)) -> c_1(f#(g(x,s(y)))):1
2:W:f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
-->_2 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_1 f#(g(s(x),y)) -> c_1(f#(g(x,s(y)))):1
3:W:h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
-->_1 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 h#(i(x,y)) -> c_6(h#(h(y)),h#(y)):4
-->_1 h#(g(x,s(y))) -> c_5(h#(g(s(x),y))):3
4:W:h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
-->_2 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_2 h#(i(x,y)) -> c_6(h#(h(y)),h#(y)):4
-->_1 h#(i(x,y)) -> c_6(h#(h(y)),h#(y)):4
-->_2 h#(g(x,s(y))) -> c_5(h#(g(s(x),y))):3
-->_1 h#(g(x,s(y))) -> c_5(h#(g(s(x),y))):3
5:W:h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
-->_1 h#(s(f(x))) -> c_7(h#(f(x)),f#(x)):5
-->_1 h#(i(x,y)) -> c_6(h#(h(y)),h#(y)):4
-->_1 h#(g(x,s(y))) -> c_5(h#(g(s(x),y))):3
-->_2 f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x))):2
-->_2 f#(g(s(x),y)) -> c_1(f#(g(x,s(y)))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(g(s(x),y)) -> c_1(f#(g(x,s(y))))
5: h#(s(f(x))) -> c_7(h#(f(x)),f#(x))
4: h#(i(x,y)) -> c_6(h#(h(y)),h#(y))
3: h#(g(x,s(y))) -> c_5(h#(g(s(x),y)))
2: f#(s(x)) -> c_2(f#(h(s(x))),h#(s(x)))
****** Step 7.b:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(g(s(x),y)) -> f(g(x,s(y)))
f(s(x)) -> s(s(f(h(s(x)))))
g(x,x) -> g(a(),b())
h(g(x,s(y))) -> h(g(s(x),y))
h(i(x,y)) -> i(i(c(),h(h(y))),x)
h(s(f(x))) -> h(f(x))
i(x,x) -> i(a(),b())
- Signature:
{f/1,g/2,h/1,i/2,f#/1,g#/2,h#/1,i#/2} / {a/0,b/0,c/0,s/1,c_1/1,c_2/2,c_3/1,c_4/6,c_5/1,c_6/2,c_7/2,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#,i#} and constructors {a,b,c,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))