```* Step 1: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
=(x,y) -> xor(x,xor(y,true()))
implies(x,y) -> xor(and(x,y),xor(x,true()))
not(x) -> xor(x,true())
or(x,y) -> xor(and(x,y),xor(x,y))
- Signature:
{=/2,implies/2,not/1,or/2} / {and/2,true/0,xor/2}
- Obligation:
runtime complexity wrt. defined symbols {=,implies,not,or} and constructors {and,true,xor}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak dependency pairs:

Strict DPs
=#(x,y) -> c_1(x,y)
implies#(x,y) -> c_2(x,y,x)
not#(x) -> c_3(x)
or#(x,y) -> c_4(x,y,x,y)
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
=#(x,y) -> c_1(x,y)
implies#(x,y) -> c_2(x,y,x)
not#(x) -> c_3(x)
or#(x,y) -> c_4(x,y,x,y)
- Strict TRS:
=(x,y) -> xor(x,xor(y,true()))
implies(x,y) -> xor(and(x,y),xor(x,true()))
not(x) -> xor(x,true())
or(x,y) -> xor(and(x,y),xor(x,y))
- Signature:
{=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
- Obligation:
runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
=#(x,y) -> c_1(x,y)
implies#(x,y) -> c_2(x,y,x)
not#(x) -> c_3(x)
or#(x,y) -> c_4(x,y,x,y)
* Step 3: PredecessorEstimationCP WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
=#(x,y) -> c_1(x,y)
implies#(x,y) -> c_2(x,y,x)
not#(x) -> c_3(x)
or#(x,y) -> c_4(x,y,x,y)
- Signature:
{=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
- Obligation:
runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: =#(x,y) -> c_1(x,y)
2: implies#(x,y) -> c_2(x,y,x)
3: not#(x) -> c_3(x)
4: or#(x,y) -> c_4(x,y,x,y)

The strictly oriented rules are moved into the weak component.
** Step 3.a:1: NaturalMI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
=#(x,y) -> c_1(x,y)
implies#(x,y) -> c_2(x,y,x)
not#(x) -> c_3(x)
or#(x,y) -> c_4(x,y,x,y)
- Signature:
{=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
- Obligation:
runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, 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 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
none

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(=) = [1] x1 + [0]
p(and) = [0]
p(implies) = [0]
p(not) = [0]
p(or) = [0]
p(true) = [0]
p(xor) = [0]
p(=#) = [1]
p(implies#) = [5] x1 + [1]
p(not#) = [1]
p(or#) = [1]
p(c_1) = [0]
p(c_2) = [5] x1 + [0]
p(c_3) = [0]
p(c_4) = [0]

Following rules are strictly oriented:
=#(x,y) = [1]
> [0]
= c_1(x,y)

implies#(x,y) = [5] x + [1]
> [5] x + [0]
= c_2(x,y,x)

not#(x) = [1]
> [0]
= c_3(x)

or#(x,y) = [1]
> [0]
= c_4(x,y,x,y)

Following rules are (at-least) weakly oriented:

** Step 3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
=#(x,y) -> c_1(x,y)
implies#(x,y) -> c_2(x,y,x)
not#(x) -> c_3(x)
or#(x,y) -> c_4(x,y,x,y)
- Signature:
{=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
- Obligation:
runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
=#(x,y) -> c_1(x,y)
implies#(x,y) -> c_2(x,y,x)
not#(x) -> c_3(x)
or#(x,y) -> c_4(x,y,x,y)
- Signature:
{=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
- Obligation:
runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:=#(x,y) -> c_1(x,y)
-->_2 or#(x,y) -> c_4(x,y,x,y):4
-->_1 or#(x,y) -> c_4(x,y,x,y):4
-->_2 not#(x) -> c_3(x):3
-->_1 not#(x) -> c_3(x):3
-->_2 implies#(x,y) -> c_2(x,y,x):2
-->_1 implies#(x,y) -> c_2(x,y,x):2
-->_2 =#(x,y) -> c_1(x,y):1
-->_1 =#(x,y) -> c_1(x,y):1

2:W:implies#(x,y) -> c_2(x,y,x)
-->_3 or#(x,y) -> c_4(x,y,x,y):4
-->_2 or#(x,y) -> c_4(x,y,x,y):4
-->_1 or#(x,y) -> c_4(x,y,x,y):4
-->_3 not#(x) -> c_3(x):3
-->_2 not#(x) -> c_3(x):3
-->_1 not#(x) -> c_3(x):3
-->_3 implies#(x,y) -> c_2(x,y,x):2
-->_2 implies#(x,y) -> c_2(x,y,x):2
-->_1 implies#(x,y) -> c_2(x,y,x):2
-->_3 =#(x,y) -> c_1(x,y):1
-->_2 =#(x,y) -> c_1(x,y):1
-->_1 =#(x,y) -> c_1(x,y):1

3:W:not#(x) -> c_3(x)
-->_1 or#(x,y) -> c_4(x,y,x,y):4
-->_1 not#(x) -> c_3(x):3
-->_1 implies#(x,y) -> c_2(x,y,x):2
-->_1 =#(x,y) -> c_1(x,y):1

4:W:or#(x,y) -> c_4(x,y,x,y)
-->_4 or#(x,y) -> c_4(x,y,x,y):4
-->_3 or#(x,y) -> c_4(x,y,x,y):4
-->_2 or#(x,y) -> c_4(x,y,x,y):4
-->_1 or#(x,y) -> c_4(x,y,x,y):4
-->_4 not#(x) -> c_3(x):3
-->_3 not#(x) -> c_3(x):3
-->_2 not#(x) -> c_3(x):3
-->_1 not#(x) -> c_3(x):3
-->_4 implies#(x,y) -> c_2(x,y,x):2
-->_3 implies#(x,y) -> c_2(x,y,x):2
-->_2 implies#(x,y) -> c_2(x,y,x):2
-->_1 implies#(x,y) -> c_2(x,y,x):2
-->_4 =#(x,y) -> c_1(x,y):1
-->_3 =#(x,y) -> c_1(x,y):1
-->_2 =#(x,y) -> c_1(x,y):1
-->_1 =#(x,y) -> 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,y) -> c_1(x,y)
4: or#(x,y) -> c_4(x,y,x,y)
3: not#(x) -> c_3(x)
2: implies#(x,y) -> c_2(x,y,x)
** Step 3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
- Obligation:
runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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