* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
+ Considered Problem:
- Strict TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
f(g(f(x))) -> f(h(s(0()),x))
f(g(h(x,y))) -> f(h(s(x),y))
f(h(x,h(y,z))) -> f(h(+(x,y),z))
- Signature:
{+/2,f/1} / {0/0,g/1,h/2,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,f} and constructors {0,g,h,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
f(g(f(x))) -> f(h(s(0()),x))
f(g(h(x,y))) -> f(h(s(x),y))
f(h(x,h(y,z))) -> f(h(+(x,y),z))
- Signature:
{+/2,f/1} / {0/0,g/1,h/2,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,f} and constructors {0,g,h,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
+(x,y){y -> s(y)} =
+(x,s(y)) ->^+ s(+(x,y))
= C[+(x,y) = +(x,y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
f(g(f(x))) -> f(h(s(0()),x))
f(g(h(x,y))) -> f(h(s(x),y))
f(h(x,h(y,z))) -> f(h(+(x,y),z))
- Signature:
{+/2,f/1} / {0/0,g/1,h/2,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,f} and constructors {0,g,h,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,0()) -> c_2()
+#(x,s(y)) -> c_3(+#(x,y))
+#(0(),y) -> c_4()
+#(s(x),y) -> c_5(+#(x,y))
f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,0()) -> c_2()
+#(x,s(y)) -> c_3(+#(x,y))
+#(0(),y) -> c_4()
+#(s(x),y) -> c_5(+#(x,y))
f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
f(g(f(x))) -> f(h(s(0()),x))
f(g(h(x,y))) -> f(h(s(x),y))
f(h(x,h(y,z))) -> f(h(+(x,y),z))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,0()) -> c_2()
+#(x,s(y)) -> c_3(+#(x,y))
+#(0(),y) -> c_4()
+#(s(x),y) -> c_5(+#(x,y))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,0()) -> c_2()
+#(x,s(y)) -> c_3(+#(x,y))
+#(0(),y) -> c_4()
+#(s(x),y) -> c_5(+#(x,y))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4}
by application of
Pre({2,4}) = {1,3,5,7}.
Here rules are labelled as follows:
1: +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
2: +#(x,0()) -> c_2()
3: +#(x,s(y)) -> c_3(+#(x,y))
4: +#(0(),y) -> c_4()
5: +#(s(x),y) -> c_5(+#(x,y))
6: f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
7: f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak DPs:
+#(x,0()) -> c_2()
+#(0(),y) -> c_4()
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(0(),y) -> c_4():7
-->_1 +#(0(),y) -> c_4():7
-->_2 +#(x,0()) -> c_2():6
-->_1 +#(x,0()) -> c_2():6
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
2:S:+#(x,s(y)) -> c_3(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(0(),y) -> c_4():7
-->_1 +#(x,0()) -> c_2():6
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
3:S:+#(s(x),y) -> c_5(+#(x,y))
-->_1 +#(0(),y) -> c_4():7
-->_1 +#(x,0()) -> c_2():6
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
4:S:f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):5
5:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_2 +#(0(),y) -> c_4():7
-->_2 +#(x,0()) -> c_2():6
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):5
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
6:W:+#(x,0()) -> c_2()
7:W:+#(0(),y) -> c_4()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: +#(x,0()) -> c_2()
7: +#(0(),y) -> c_4()
** Step 1.b:5: RemoveHeads WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
2:S:+#(x,s(y)) -> c_3(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
3:S:+#(s(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
4:S:f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):5
5:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):5
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(4,f#(g(h(x,y))) -> c_7(f#(h(s(x),y))))]
** Step 1.b:6: Decompose WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
Problem (S)
- Strict DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
*** Step 1.b:6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,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: +#(x,s(y)) -> c_3(+#(x,y))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,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) = {1,2},
uargs(c_3) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{+,+#,f#}
TcT has computed the following interpretation:
p(+) = [1 0 0] [1 0 1] [1]
[0 1 0] x1 + [0 1 0] x2 + [0]
[0 0 1] [0 1 1] [0]
p(0) = [0]
[1]
[0]
p(f) = [0]
[0]
[0]
p(g) = [0]
[0]
[0]
p(h) = [0 0 0] [0 1 0] [1]
[0 1 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(s) = [0 0 0] [0]
[0 1 0] x1 + [1]
[0 0 0] [1]
p(+#) = [0 1 0] [0]
[0 0 0] x2 + [1]
[1 0 0] [1]
p(f#) = [1 1 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [1]
p(c_1) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 0] [1]
p(c_2) = [0]
[0]
[0]
p(c_3) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(c_6) = [0]
[0]
[0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [1 0 0] [1 0 0] [0]
[0 0 1] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
Following rules are strictly oriented:
+#(x,s(y)) = [0 1 0] [1]
[0 0 0] y + [1]
[0 0 0] [1]
> [0 1 0] [0]
[0 0 0] y + [0]
[0 0 0] [0]
= c_3(+#(x,y))
Following rules are (at-least) weakly oriented:
+#(x,+(y,z)) = [0 1 0] [0 1 0] [0]
[0 0 0] y + [0 0 0] z + [1]
[1 0 0] [1 0 1] [2]
>= [0 1 0] [0 1 0] [0]
[0 0 0] y + [0 0 0] z + [0]
[0 0 0] [1 0 0] [2]
= c_1(+#(+(x,y),z),+#(x,y))
+#(s(x),y) = [0 1 0] [0]
[0 0 0] y + [1]
[1 0 0] [1]
>= [0 1 0] [0]
[0 0 0] y + [0]
[1 0 0] [1]
= c_5(+#(x,y))
f#(h(x,h(y,z))) = [0 1 0] [0 2 0] [0 2 0] [1]
[0 0 0] x + [0 0 0] y + [0 0 0] z + [1]
[0 0 0] [0 0 0] [0 0 0] [1]
>= [0 1 0] [0 2 0] [0 2 0] [1]
[0 0 0] x + [0 0 0] y + [0 0 0] z + [1]
[0 0 0] [0 0 0] [0 0 0] [1]
= c_8(f#(h(+(x,y),z)),+#(x,y))
+(x,+(y,z)) = [1 0 0] [1 0 1] [1 1 2] [2]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 1] [0 1 1] [0 2 1] [0]
>= [1 0 0] [1 0 1] [1 0 1] [2]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 1] [0 1 1] [0 1 1] [0]
= +(+(x,y),z)
+(x,0()) = [1 0 0] [1]
[0 1 0] x + [1]
[0 0 1] [1]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
+(x,s(y)) = [1 0 0] [0 0 0] [2]
[0 1 0] x + [0 1 0] y + [1]
[0 0 1] [0 1 0] [2]
>= [0 0 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [1]
[0 0 0] [0 0 0] [1]
= s(+(x,y))
+(0(),y) = [1 0 1] [1]
[0 1 0] y + [1]
[0 1 1] [0]
>= [1 0 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= y
+(s(x),y) = [0 0 0] [1 0 1] [1]
[0 1 0] x + [0 1 0] y + [1]
[0 0 0] [0 1 1] [1]
>= [0 0 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [1]
[0 0 0] [0 0 0] [1]
= s(+(x,y))
**** Step 1.b:6.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,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:
1: +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,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) = {1,2},
uargs(c_3) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{+,+#,f#}
TcT has computed the following interpretation:
p(+) = [1 0 0] [1 0 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0]
[0 0 1] [0 0 1] [1]
p(0) = [0]
[0]
[1]
p(f) = [0]
[0]
[0]
p(g) = [0]
[0]
[0]
p(h) = [0 0 0] [0 0 0] [1]
[0 0 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(s) = [0 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(+#) = [0 0 1] [0]
[0 0 0] x2 + [0]
[0 1 0] [0]
p(f#) = [0 1 0] [0]
[0 0 0] x1 + [0]
[1 0 0] [0]
p(c_1) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(c_6) = [0]
[0]
[0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
Following rules are strictly oriented:
+#(x,+(y,z)) = [0 0 1] [0 0 1] [1]
[0 0 0] y + [0 0 0] z + [0]
[0 1 0] [0 1 0] [0]
> [0 0 1] [0 0 1] [0]
[0 0 0] y + [0 0 0] z + [0]
[0 0 0] [0 0 0] [0]
= c_1(+#(+(x,y),z),+#(x,y))
Following rules are (at-least) weakly oriented:
+#(x,s(y)) = [0 0 1] [0]
[0 0 0] y + [0]
[0 0 0] [0]
>= [0 0 1] [0]
[0 0 0] y + [0]
[0 0 0] [0]
= c_3(+#(x,y))
+#(s(x),y) = [0 0 1] [0]
[0 0 0] y + [0]
[0 1 0] [0]
>= [0 0 1] [0]
[0 0 0] y + [0]
[0 1 0] [0]
= c_5(+#(x,y))
f#(h(x,h(y,z))) = [0 0 1] [0 0 1] [0]
[0 0 0] y + [0 0 0] z + [0]
[0 0 0] [0 0 0] [1]
>= [0 0 1] [0 0 1] [0]
[0 0 0] y + [0 0 0] z + [0]
[0 0 0] [0 0 0] [0]
= c_8(f#(h(+(x,y),z)),+#(x,y))
+(x,+(y,z)) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 1] [0 0 1] [0 0 1] [2]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 1] [0 0 1] [0 0 1] [2]
= +(+(x,y),z)
+(x,0()) = [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [2]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
+(x,s(y)) = [1 0 0] [0 0 0] [0]
[0 1 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
>= [0 0 0] [0 0 0] [0]
[0 0 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
= s(+(x,y))
+(0(),y) = [1 0 0] [0]
[0 1 0] y + [0]
[0 0 1] [2]
>= [1 0 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= y
+(s(x),y) = [0 0 0] [1 0 0] [0]
[0 0 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [1]
>= [0 0 0] [0 0 0] [0]
[0 0 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
= s(+(x,y))
***** Step 1.b:6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:1.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
and a lower component
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Further, following extension rules are added to the lower component.
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
****** Step 1.b:6.a:1.b:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
The strictly oriented rules are moved into the weak component.
******* Step 1.b:6.a:1.b:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
NaturalMI {miDimension = 1, 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:
The following argument positions are considered usable:
uargs(c_8) = {1}
Following symbols are considered usable:
{+,+#,f#}
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [1]
p(0) = [1]
p(f) = [1] x1 + [0]
p(g) = [1]
p(h) = [1] x1 + [1] x2 + [3]
p(s) = [0]
p(+#) = [0]
p(f#) = [2] x1 + [0]
p(c_1) = [2]
p(c_2) = [8]
p(c_3) = [4] x1 + [1]
p(c_4) = [4]
p(c_5) = [2] x1 + [2]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
Following rules are strictly oriented:
f#(h(x,h(y,z))) = [2] x + [2] y + [2] z + [12]
> [2] x + [2] y + [2] z + [8]
= c_8(f#(h(+(x,y),z)),+#(x,y))
Following rules are (at-least) weakly oriented:
+(x,+(y,z)) = [1] x + [1] y + [1] z + [2]
>= [1] x + [1] y + [1] z + [2]
= +(+(x,y),z)
+(x,0()) = [1] x + [2]
>= [1] x + [0]
= x
+(x,s(y)) = [1] x + [1]
>= [0]
= s(+(x,y))
+(0(),y) = [1] y + [2]
>= [1] y + [0]
= y
+(s(x),y) = [1] y + [1]
>= [0]
= s(+(x,y))
******* Step 1.b:6.a:1.b:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
******* Step 1.b:6.a:1.b:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
******* Step 1.b:6.a:1.b:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
****** Step 1.b:6.a:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: +#(s(x),y) -> c_5(+#(x,y))
The strictly oriented rules are moved into the weak component.
******* Step 1.b:6.a:1.b:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
+#(s(x),y) -> c_5(+#(x,y))
- Weak DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_1) = {1,2},
uargs(c_3) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
{+,+#,f#}
TcT has computed the following interpretation:
p(+) = 1 + x1 + x2
p(0) = 1
p(f) = 1
p(g) = 1
p(h) = 1 + x1 + x2
p(s) = 1 + x1
p(+#) = 1 + 2*x1 + 2*x1*x2 + 2*x2 + x2^2
p(f#) = 1 + x1 + x1^2
p(c_1) = x1 + x2
p(c_2) = 0
p(c_3) = x1
p(c_4) = 0
p(c_5) = x1
p(c_6) = 0
p(c_7) = 0
p(c_8) = x2
Following rules are strictly oriented:
+#(s(x),y) = 3 + 2*x + 2*x*y + 4*y + y^2
> 1 + 2*x + 2*x*y + 2*y + y^2
= c_5(+#(x,y))
Following rules are (at-least) weakly oriented:
+#(x,+(y,z)) = 4 + 4*x + 2*x*y + 2*x*z + 4*y + 2*y*z + y^2 + 4*z + z^2
>= 4 + 4*x + 2*x*y + 2*x*z + 4*y + 2*y*z + y^2 + 4*z + z^2
= c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) = 4 + 4*x + 2*x*y + 4*y + y^2
>= 1 + 2*x + 2*x*y + 2*y + y^2
= c_3(+#(x,y))
f#(h(x,h(y,z))) = 7 + 5*x + 2*x*y + 2*x*z + x^2 + 5*y + 2*y*z + y^2 + 5*z + z^2
>= 1 + 2*x + 2*x*y + 2*y + y^2
= +#(x,y)
f#(h(x,h(y,z))) = 7 + 5*x + 2*x*y + 2*x*z + x^2 + 5*y + 2*y*z + y^2 + 5*z + z^2
>= 7 + 5*x + 2*x*y + 2*x*z + x^2 + 5*y + 2*y*z + y^2 + 5*z + z^2
= f#(h(+(x,y),z))
+(x,+(y,z)) = 2 + x + y + z
>= 2 + x + y + z
= +(+(x,y),z)
+(x,0()) = 2 + x
>= x
= x
+(x,s(y)) = 2 + x + y
>= 2 + x + y
= s(+(x,y))
+(0(),y) = 2 + y
>= y
= y
+(s(x),y) = 2 + x + y
>= 2 + x + y
= s(+(x,y))
******* Step 1.b:6.a:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
******* Step 1.b:6.a:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
2:W:+#(x,s(y)) -> c_3(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
3:W:+#(s(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
4:W:f#(h(x,h(y,z))) -> +#(x,y)
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
5:W:f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
-->_1 f#(h(x,h(y,z))) -> f#(h(+(x,y),z)):5
-->_1 f#(h(x,h(y,z))) -> +#(x,y):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
4: f#(h(x,h(y,z))) -> +#(x,y)
1: +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
3: +#(s(x),y) -> c_5(+#(x,y))
2: +#(x,s(y)) -> c_3(+#(x,y))
******* Step 1.b:6.a:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak DPs:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):4
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):1
2:W:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):4
-->_1 +#(s(x),y) -> c_5(+#(x,y)):4
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
3:W:+#(x,s(y)) -> c_3(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):4
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
4:W:+#(s(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):4
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: +#(s(x),y) -> c_5(+#(x,y))
3: +#(x,s(y)) -> c_3(+#(x,y))
2: +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
*** Step 1.b:6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
NaturalMI {miDimension = 1, 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:
The following argument positions are considered usable:
uargs(c_8) = {1}
Following symbols are considered usable:
{+#,f#}
TcT has computed the following interpretation:
p(+) = [6]
p(0) = [0]
p(f) = [1]
p(g) = [1] x1 + [0]
p(h) = [1] x2 + [2]
p(s) = [1] x1 + [2]
p(+#) = [1] x2 + [1]
p(f#) = [4] x1 + [1]
p(c_1) = [1] x2 + [1]
p(c_2) = [0]
p(c_3) = [8] x1 + [0]
p(c_4) = [2]
p(c_5) = [8] x1 + [1]
p(c_6) = [1] x1 + [2]
p(c_7) = [1]
p(c_8) = [1] x1 + [6]
Following rules are strictly oriented:
f#(h(x,h(y,z))) = [4] z + [17]
> [4] z + [15]
= c_8(f#(h(+(x,y),z)))
Following rules are (at-least) weakly oriented:
**** Step 1.b:6.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
**** Step 1.b:6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
- Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^3))