```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
++(x,g(y,z)) -> g(++(x,y),z)
++(x,nil()) -> x
f(x,g(y,z)) -> g(f(x,y),z)
f(x,nil()) -> g(nil(),x)
max(g(g(g(x,y),z),u())) -> max'(max(g(g(x,y),z)),u())
max(g(g(nil(),x),y)) -> max'(x,y)
mem(g(x,y),z) -> or(=(y,z),mem(x,z))
mem(nil(),y) -> false()
null(g(x,y)) -> false()
null(nil()) -> true()
- Signature:
{++/2,f/2,max/1,mem/2,null/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0,u/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++,f,max,mem,null} and constructors {=,false,g,max',nil
,or,true,u}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
++#(x,g(y,z)) -> c_1(++#(x,y))
++#(x,nil()) -> c_2()
f#(x,g(y,z)) -> c_3(f#(x,y))
f#(x,nil()) -> c_4()
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
max#(g(g(nil(),x),y)) -> c_6()
mem#(g(x,y),z) -> c_7(mem#(x,z))
mem#(nil(),y) -> c_8()
null#(g(x,y)) -> c_9()
null#(nil()) -> c_10()
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
++#(x,nil()) -> c_2()
f#(x,g(y,z)) -> c_3(f#(x,y))
f#(x,nil()) -> c_4()
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
max#(g(g(nil(),x),y)) -> c_6()
mem#(g(x,y),z) -> c_7(mem#(x,z))
mem#(nil(),y) -> c_8()
null#(g(x,y)) -> c_9()
null#(nil()) -> c_10()
- Strict TRS:
++(x,g(y,z)) -> g(++(x,y),z)
++(x,nil()) -> x
f(x,g(y,z)) -> g(f(x,y),z)
f(x,nil()) -> g(nil(),x)
max(g(g(g(x,y),z),u())) -> max'(max(g(g(x,y),z)),u())
max(g(g(nil(),x),y)) -> max'(x,y)
mem(g(x,y),z) -> or(=(y,z),mem(x,z))
mem(nil(),y) -> false()
null(g(x,y)) -> false()
null(nil()) -> true()
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
++#(x,g(y,z)) -> c_1(++#(x,y))
++#(x,nil()) -> c_2()
f#(x,g(y,z)) -> c_3(f#(x,y))
f#(x,nil()) -> c_4()
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
max#(g(g(nil(),x),y)) -> c_6()
mem#(g(x,y),z) -> c_7(mem#(x,z))
mem#(nil(),y) -> c_8()
null#(g(x,y)) -> c_9()
null#(nil()) -> c_10()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
++#(x,nil()) -> c_2()
f#(x,g(y,z)) -> c_3(f#(x,y))
f#(x,nil()) -> c_4()
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
max#(g(g(nil(),x),y)) -> c_6()
mem#(g(x,y),z) -> c_7(mem#(x,z))
mem#(nil(),y) -> c_8()
null#(g(x,y)) -> c_9()
null#(nil()) -> c_10()
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4,6,8,9,10}
by application of
Pre({2,4,6,8,9,10}) = {1,3,5,7}.
Here rules are labelled as follows:
1: ++#(x,g(y,z)) -> c_1(++#(x,y))
2: ++#(x,nil()) -> c_2()
3: f#(x,g(y,z)) -> c_3(f#(x,y))
4: f#(x,nil()) -> c_4()
5: max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
6: max#(g(g(nil(),x),y)) -> c_6()
7: mem#(g(x,y),z) -> c_7(mem#(x,z))
8: mem#(nil(),y) -> c_8()
9: null#(g(x,y)) -> c_9()
10: null#(nil()) -> c_10()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
f#(x,g(y,z)) -> c_3(f#(x,y))
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Weak DPs:
++#(x,nil()) -> c_2()
f#(x,nil()) -> c_4()
max#(g(g(nil(),x),y)) -> c_6()
mem#(nil(),y) -> c_8()
null#(g(x,y)) -> c_9()
null#(nil()) -> c_10()
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:++#(x,g(y,z)) -> c_1(++#(x,y))
-->_1 ++#(x,nil()) -> c_2():5
-->_1 ++#(x,g(y,z)) -> c_1(++#(x,y)):1

2:S:f#(x,g(y,z)) -> c_3(f#(x,y))
-->_1 f#(x,nil()) -> c_4():6
-->_1 f#(x,g(y,z)) -> c_3(f#(x,y)):2

3:S:max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
-->_1 max#(g(g(nil(),x),y)) -> c_6():7
-->_1 max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z))):3

4:S:mem#(g(x,y),z) -> c_7(mem#(x,z))
-->_1 mem#(nil(),y) -> c_8():8
-->_1 mem#(g(x,y),z) -> c_7(mem#(x,z)):4

5:W:++#(x,nil()) -> c_2()

6:W:f#(x,nil()) -> c_4()

7:W:max#(g(g(nil(),x),y)) -> c_6()

8:W:mem#(nil(),y) -> c_8()

9:W:null#(g(x,y)) -> c_9()

10:W:null#(nil()) -> c_10()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: null#(nil()) -> c_10()
9: null#(g(x,y)) -> c_9()
8: mem#(nil(),y) -> c_8()
7: max#(g(g(nil(),x),y)) -> c_6()
6: f#(x,nil()) -> c_4()
5: ++#(x,nil()) -> c_2()
* Step 5: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
f#(x,g(y,z)) -> c_3(f#(x,y))
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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,g(y,z)) -> c_1(++#(x,y))
- Weak DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g
,max',nil,or,true,u}

Problem (S)
- Strict DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Weak DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g
,max',nil,or,true,u}
** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
- Weak DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:++#(x,g(y,z)) -> c_1(++#(x,y))
-->_1 ++#(x,g(y,z)) -> c_1(++#(x,y)):1

2:W:f#(x,g(y,z)) -> c_3(f#(x,y))
-->_1 f#(x,g(y,z)) -> c_3(f#(x,y)):2

3:W:max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
-->_1 max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z))):3

4:W:mem#(g(x,y),z) -> c_7(mem#(x,z))
-->_1 mem#(g(x,y),z) -> c_7(mem#(x,z)):4

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: mem#(g(x,y),z) -> c_7(mem#(x,z))
3: max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
2: f#(x,g(y,z)) -> c_3(f#(x,y))
** Step 5.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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: ++#(x,g(y,z)) -> c_1(++#(x,y))

The strictly oriented rules are moved into the weak component.
*** Step 5.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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_1) = {1}

Following symbols are considered usable:
{++#,f#,max#,mem#,null#}
TcT has computed the following interpretation:
p(++) = [0]
p(=) = [0]
p(f) = [1] x1 + [1] x2 + [1]
p(false) = [2]
p(g) = [1] x1 + [8]
p(max) = [1]
p(max') = [0]
p(mem) = [2]
p(nil) = [2]
p(null) = [1] x1 + [1]
p(or) = [4]
p(true) = [1]
p(u) = [0]
p(++#) = [2] x1 + [2] x2 + [0]
p(f#) = [2] x2 + [1]
p(max#) = [8] x1 + [1]
p(mem#) = [2] x1 + [8] x2 + [0]
p(null#) = [8] x1 + [1]
p(c_1) = [1] x1 + [15]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [1] x1 + [1]
p(c_8) = [1]
p(c_9) = [0]
p(c_10) = [0]

Following rules are strictly oriented:
++#(x,g(y,z)) = [2] x + [2] y + [16]
> [2] x + [2] y + [15]
= c_1(++#(x,y))

Following rules are (at-least) weakly oriented:

*** Step 5.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 5.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:++#(x,g(y,z)) -> c_1(++#(x,y))
-->_1 ++#(x,g(y,z)) -> c_1(++#(x,y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: ++#(x,g(y,z)) -> c_1(++#(x,y))
*** Step 5.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Weak DPs:
++#(x,g(y,z)) -> c_1(++#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(x,g(y,z)) -> c_3(f#(x,y))
-->_1 f#(x,g(y,z)) -> c_3(f#(x,y)):1

2:S:max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
-->_1 max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z))):2

3:S:mem#(g(x,y),z) -> c_7(mem#(x,z))
-->_1 mem#(g(x,y),z) -> c_7(mem#(x,z)):3

4:W:++#(x,g(y,z)) -> c_1(++#(x,y))
-->_1 ++#(x,g(y,z)) -> c_1(++#(x,y)):4

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: ++#(x,g(y,z)) -> c_1(++#(x,y))
** Step 5.b:2: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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:
f#(x,g(y,z)) -> c_3(f#(x,y))
- Weak DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g
,max',nil,or,true,u}

Problem (S)
- Strict DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Weak DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g
,max',nil,or,true,u}
*** Step 5.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
- Weak DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(x,g(y,z)) -> c_3(f#(x,y))
-->_1 f#(x,g(y,z)) -> c_3(f#(x,y)):1

2:W:max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
-->_1 max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z))):2

3:W:mem#(g(x,y),z) -> c_7(mem#(x,z))
-->_1 mem#(g(x,y),z) -> c_7(mem#(x,z)):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: mem#(g(x,y),z) -> c_7(mem#(x,z))
2: max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
*** Step 5.b:2.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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#(x,g(y,z)) -> c_3(f#(x,y))

The strictly oriented rules are moved into the weak component.
**** Step 5.b:2.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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_3) = {1}

Following symbols are considered usable:
{++#,f#,max#,mem#,null#}
TcT has computed the following interpretation:
p(++) = [0]
p(=) = [1] x1 + [1] x2 + [0]
p(f) = [0]
p(false) = [0]
p(g) = [1] x1 + [1]
p(max) = [8]
p(max') = [1]
p(mem) = [1] x1 + [1]
p(nil) = [0]
p(null) = [1] x1 + [1]
p(or) = [1] x1 + [1]
p(true) = [0]
p(u) = [1]
p(++#) = [1] x1 + [1] x2 + [1]
p(f#) = [1] x1 + [8] x2 + [1]
p(max#) = [1] x1 + [1]
p(mem#) = [1]
p(null#) = [0]
p(c_1) = [1] x1 + [2]
p(c_2) = [1]
p(c_3) = [1] x1 + [5]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [0]
p(c_7) = [1] x1 + [2]
p(c_8) = [2]
p(c_9) = [2]
p(c_10) = [8]

Following rules are strictly oriented:
f#(x,g(y,z)) = [1] x + [8] y + [9]
> [1] x + [8] y + [6]
= c_3(f#(x,y))

Following rules are (at-least) weakly oriented:

**** Step 5.b:2.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 5.b:2.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(x,g(y,z)) -> c_3(f#(x,y))
-->_1 f#(x,g(y,z)) -> c_3(f#(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#(x,g(y,z)) -> c_3(f#(x,y))
**** Step 5.b:2.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

*** Step 5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Weak DPs:
f#(x,g(y,z)) -> c_3(f#(x,y))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
-->_1 max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z))):1

2:S:mem#(g(x,y),z) -> c_7(mem#(x,z))
-->_1 mem#(g(x,y),z) -> c_7(mem#(x,z)):2

3:W:f#(x,g(y,z)) -> c_3(f#(x,y))
-->_1 f#(x,g(y,z)) -> c_3(f#(x,y)):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: f#(x,g(y,z)) -> c_3(f#(x,y))
*** Step 5.b:2.b:2: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
- Weak DPs:
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g
,max',nil,or,true,u}

Problem (S)
- Strict DPs:
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Weak DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g
,max',nil,or,true,u}
**** Step 5.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
- Weak DPs:
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
-->_1 max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z))):1

2:W:mem#(g(x,y),z) -> c_7(mem#(x,z))
-->_1 mem#(g(x,y),z) -> c_7(mem#(x,z)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: mem#(g(x,y),z) -> c_7(mem#(x,z))
**** Step 5.b:2.b:2.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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: max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))

The strictly oriented rules are moved into the weak component.
***** Step 5.b:2.b:2.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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_5) = {1}

Following symbols are considered usable:
{++#,f#,max#,mem#,null#}
TcT has computed the following interpretation:
p(++) = [1] x2 + [0]
p(=) = [0]
p(f) = [1] x1 + [8] x2 + [4]
p(false) = [1]
p(g) = [1] x1 + [1] x2 + [0]
p(max) = [1] x1 + [4]
p(max') = [1] x2 + [1]
p(mem) = [2] x2 + [0]
p(nil) = [0]
p(null) = [0]
p(or) = [0]
p(true) = [4]
p(u) = [2]
p(++#) = [1] x1 + [0]
p(f#) = [1] x2 + [2]
p(max#) = [8] x1 + [4]
p(mem#) = [8] x1 + [1] x2 + [1]
p(null#) = [2]
p(c_1) = [2]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [2]
p(c_5) = [1] x1 + [6]
p(c_6) = [0]
p(c_7) = [2]
p(c_8) = [0]
p(c_9) = [1]
p(c_10) = [1]

Following rules are strictly oriented:
max#(g(g(g(x,y),z),u())) = [8] x + [8] y + [8] z + [20]
> [8] x + [8] y + [8] z + [10]
= c_5(max#(g(g(x,y),z)))

Following rules are (at-least) weakly oriented:

***** Step 5.b:2.b:2.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 5.b:2.b:2.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
-->_1 max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(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: max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
***** Step 5.b:2.b:2.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

**** Step 5.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Weak DPs:
max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:mem#(g(x,y),z) -> c_7(mem#(x,z))
-->_1 mem#(g(x,y),z) -> c_7(mem#(x,z)):1

2:W:max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
-->_1 max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z))):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: max#(g(g(g(x,y),z),u())) -> c_5(max#(g(g(x,y),z)))
**** Step 5.b:2.b:2.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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: mem#(g(x,y),z) -> c_7(mem#(x,z))

The strictly oriented rules are moved into the weak component.
***** Step 5.b:2.b:2.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ 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_7) = {1}

Following symbols are considered usable:
{++#,f#,max#,mem#,null#}
TcT has computed the following interpretation:
p(++) = [0]
p(=) = [1] x1 + [1] x2 + [0]
p(f) = [0]
p(false) = [0]
p(g) = [1] x1 + [1] x2 + [4]
p(max) = [0]
p(max') = [1] x1 + [1] x2 + [0]
p(mem) = [0]
p(nil) = [0]
p(null) = [0]
p(or) = [1] x1 + [1] x2 + [0]
p(true) = [0]
p(u) = [0]
p(++#) = [1] x1 + [8] x2 + [0]
p(f#) = [8]
p(max#) = [8] x1 + [1]
p(mem#) = [4] x1 + [8]
p(null#) = [2] x1 + [1]
p(c_1) = [4] x1 + [1]
p(c_2) = [1]
p(c_3) = [2] x1 + [1]
p(c_4) = [0]
p(c_5) = [4] x1 + [8]
p(c_6) = [1]
p(c_7) = [1] x1 + [8]
p(c_8) = [0]
p(c_9) = [1]
p(c_10) = [0]

Following rules are strictly oriented:
mem#(g(x,y),z) = [4] x + [4] y + [24]
> [4] x + [16]
= c_7(mem#(x,z))

Following rules are (at-least) weakly oriented:

***** Step 5.b:2.b:2.b:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 5.b:2.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mem#(g(x,y),z) -> c_7(mem#(x,z))
- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:mem#(g(x,y),z) -> c_7(mem#(x,z))
-->_1 mem#(g(x,y),z) -> c_7(mem#(x,z)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: mem#(g(x,y),z) -> c_7(mem#(x,z))
***** Step 5.b:2.b:2.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{++/2,f/2,max/1,mem/2,null/1,++#/2,f#/2,max#/1,mem#/2,null#/1} / {=/2,false/0,g/2,max'/2,nil/0,or/2,true/0
,u/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {++#,f#,max#,mem#,null#} and constructors {=,false,g,max'
,nil,or,true,u}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```