```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
circ(s,id()) -> s
circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
circ(id(),s) -> s
msubst(a,id()) -> a
subst(a,id()) -> a
- Signature:
{circ/2,msubst/2,subst/2} / {cons/2,id/0,lift/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ,msubst,subst} and constructors {cons,id,lift}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
circ#(s,id()) -> c_1()
circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
circ#(id(),s) -> c_5()
msubst#(a,id()) -> c_6()
subst#(a,id()) -> c_7()
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
circ#(s,id()) -> c_1()
circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
circ#(id(),s) -> c_5()
msubst#(a,id()) -> c_6()
subst#(a,id()) -> c_7()
- Strict TRS:
circ(s,id()) -> s
circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
circ(id(),s) -> s
msubst(a,id()) -> a
subst(a,id()) -> a
- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
circ#(s,id()) -> c_1()
circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
circ#(id(),s) -> c_5()
msubst#(a,id()) -> c_6()
subst#(a,id()) -> c_7()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
circ#(s,id()) -> c_1()
circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
circ#(id(),s) -> c_5()
msubst#(a,id()) -> c_6()
subst#(a,id()) -> c_7()
- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,5,6,7}
by application of
Pre({1,5,6,7}) = {2,3,4}.
Here rules are labelled as follows:
1: circ#(s,id()) -> c_1()
2: circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
3: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
4: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
5: circ#(id(),s) -> c_5()
6: msubst#(a,id()) -> c_6()
7: subst#(a,id()) -> c_7()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
- Weak DPs:
circ#(s,id()) -> c_1()
circ#(id(),s) -> c_5()
msubst#(a,id()) -> c_6()
subst#(a,id()) -> c_7()
- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
-->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_2 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_1 msubst#(a,id()) -> c_6():6
-->_2 circ#(id(),s) -> c_5():5
-->_2 circ#(s,id()) -> c_1():4
-->_2 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1

2:S:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
-->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_1 circ#(id(),s) -> c_5():5
-->_1 circ#(s,id()) -> c_1():4
-->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1

3:S:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
-->_1 circ#(id(),s) -> c_5():5
-->_1 circ#(s,id()) -> c_1():4
-->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1

4:W:circ#(s,id()) -> c_1()

5:W:circ#(id(),s) -> c_5()

6:W:msubst#(a,id()) -> c_6()

7:W:subst#(a,id()) -> c_7()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: subst#(a,id()) -> c_7()
6: msubst#(a,id()) -> c_6()
4: circ#(s,id()) -> c_1()
5: circ#(id(),s) -> c_5()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
-->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_2 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_2 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1

2:S:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
-->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1

3:S:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
-->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
circ#(cons(a,s),t) -> c_2(circ#(s,t))
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
circ#(cons(a,s),t) -> c_2(circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ 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: circ#(cons(a,s),t) -> c_2(circ#(s,t))
2: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
3: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))

The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
circ#(cons(a,s),t) -> c_2(circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ 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_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1}

Following symbols are considered usable:
{circ#,msubst#,subst#}
TcT has computed the following interpretation:
p(circ) = [1] x2 + [1]
p(cons) = [1] x1 + [1] x2 + [1]
p(id) = [1]
p(lift) = [3]
p(msubst) = [8] x1 + [1] x2 + [1]
p(subst) = [2] x1 + [4]
p(circ#) = [1] x1 + [1] x2 + [1]
p(msubst#) = [1] x1 + [0]
p(subst#) = [1] x2 + [0]
p(c_1) = [8]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [5]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [0]

Following rules are strictly oriented:
circ#(cons(a,s),t) = [1] a + [1] s + [1] t + [2]
> [1] s + [1] t + [1]
= c_2(circ#(s,t))

circ#(cons(lift(),s),cons(a,t)) = [1] a + [1] s + [1] t + [6]
> [1] s + [1] t + [1]
= c_3(circ#(s,t))

circ#(cons(lift(),s),cons(lift(),t)) = [1] s + [1] t + [9]
> [1] s + [1] t + [6]
= c_4(circ#(s,t))

Following rules are (at-least) weakly oriented:

** Step 6.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
circ#(cons(a,s),t) -> c_2(circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
circ#(cons(a,s),t) -> c_2(circ#(s,t))
circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:circ#(cons(a,s),t) -> c_2(circ#(s,t))
-->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1

2:W:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
-->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1

3:W:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
-->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
-->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
-->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: circ#(cons(a,s),t) -> c_2(circ#(s,t))
3: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
2: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/0,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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