* Step 1: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
e1(x1,x1,x,y,z) -> e5(x1,x,y,z)
e2(x,x,y,z,z) -> e6(x,y,z)
e2(i(),x,y,z,i()) -> e6(x,y,z)
e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z)
e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)
e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x)
e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z)
e5(i(),x,y,z) -> e6(x,y,z)
f1() -> g1()
f1() -> g2()
f2() -> g1()
f2() -> g2()
g1() -> h1()
g1() -> h2()
g2() -> h1()
g2() -> h2()
h1() -> i()
h2() -> i()
- Signature:
{e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0} / {e6/3,i/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {e1,e2,e3,e4,e5,f1,f2,g1,g2,h1,h2} and constructors {e6,i}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z))
e2#(x,x,y,z,z) -> c_2()
e2#(i(),x,y,z,i()) -> c_3()
e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_4()
e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_5(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z))
e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_6()
e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_7(e5#(x1,x,y,z))
e5#(i(),x,y,z) -> c_8()
f1#() -> c_9(g1#())
f1#() -> c_10(g2#())
f2#() -> c_11(g1#())
f2#() -> c_12(g2#())
g1#() -> c_13(h1#())
g1#() -> c_14(h2#())
g2#() -> c_15(h1#())
g2#() -> c_16(h2#())
h1#() -> c_17()
h2#() -> c_18()
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z))
e2#(x,x,y,z,z) -> c_2()
e2#(i(),x,y,z,i()) -> c_3()
e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_4()
e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_5(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z))
e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_6()
e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_7(e5#(x1,x,y,z))
e5#(i(),x,y,z) -> c_8()
f1#() -> c_9(g1#())
f1#() -> c_10(g2#())
f2#() -> c_11(g1#())
f2#() -> c_12(g2#())
g1#() -> c_13(h1#())
g1#() -> c_14(h2#())
g2#() -> c_15(h1#())
g2#() -> c_16(h2#())
h1#() -> c_17()
h2#() -> c_18()
- Weak TRS:
e1(x1,x1,x,y,z) -> e5(x1,x,y,z)
e2(x,x,y,z,z) -> e6(x,y,z)
e2(i(),x,y,z,i()) -> e6(x,y,z)
e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z)
e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)
e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x)
e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z)
e5(i(),x,y,z) -> e6(x,y,z)
f1() -> g1()
f1() -> g2()
f2() -> g1()
f2() -> g2()
g1() -> h1()
g1() -> h2()
g2() -> h1()
g2() -> h2()
h1() -> i()
h2() -> i()
- Signature:
{e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0,e1#/5,e2#/5,e3#/11,e4#/11,e5#/4,f1#/0,f2#/0,g1#/0
,g2#/0,h1#/0,h2#/0} / {e6/3,i/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1
,c_13/1,c_14/1,c_15/1,c_16/1,c_17/0,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {e1#,e2#,e3#,e4#,e5#,f1#,f2#,g1#,g2#,h1#
,h2#} and constructors {e6,i}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z))
e2#(x,x,y,z,z) -> c_2()
e2#(i(),x,y,z,i()) -> c_3()
e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_4()
e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_5(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z))
e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_6()
e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_7(e5#(x1,x,y,z))
e5#(i(),x,y,z) -> c_8()
f1#() -> c_9(g1#())
f1#() -> c_10(g2#())
f2#() -> c_11(g1#())
f2#() -> c_12(g2#())
g1#() -> c_13(h1#())
g1#() -> c_14(h2#())
g2#() -> c_15(h1#())
g2#() -> c_16(h2#())
h1#() -> c_17()
h2#() -> c_18()
* Step 3: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z))
e2#(x,x,y,z,z) -> c_2()
e2#(i(),x,y,z,i()) -> c_3()
e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_4()
e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_5(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z))
e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_6()
e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_7(e5#(x1,x,y,z))
e5#(i(),x,y,z) -> c_8()
f1#() -> c_9(g1#())
f1#() -> c_10(g2#())
f2#() -> c_11(g1#())
f2#() -> c_12(g2#())
g1#() -> c_13(h1#())
g1#() -> c_14(h2#())
g2#() -> c_15(h1#())
g2#() -> c_16(h2#())
h1#() -> c_17()
h2#() -> c_18()
- Signature:
{e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0,e1#/5,e2#/5,e3#/11,e4#/11,e5#/4,f1#/0,f2#/0,g1#/0
,g2#/0,h1#/0,h2#/0} / {e6/3,i/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1
,c_13/1,c_14/1,c_15/1,c_16/1,c_17/0,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {e1#,e2#,e3#,e4#,e5#,f1#,f2#,g1#,g2#,h1#
,h2#} and constructors {e6,i}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z))
-->_1 e5#(i(),x,y,z) -> c_8():8

2:S:e2#(x,x,y,z,z) -> c_2()

3:S:e2#(i(),x,y,z,i()) -> c_3()

4:S:e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_4()

5:S:e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_5(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z))
-->_1 e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_7(e5#(x1,x,y,z)):7
-->_1 e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_6():6

6:S:e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_6()

7:S:e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_7(e5#(x1,x,y,z))
-->_1 e5#(i(),x,y,z) -> c_8():8

8:S:e5#(i(),x,y,z) -> c_8()

9:S:f1#() -> c_9(g1#())
-->_1 g1#() -> c_14(h2#()):14
-->_1 g1#() -> c_13(h1#()):13

10:S:f1#() -> c_10(g2#())
-->_1 g2#() -> c_16(h2#()):16
-->_1 g2#() -> c_15(h1#()):15

11:S:f2#() -> c_11(g1#())
-->_1 g1#() -> c_14(h2#()):14
-->_1 g1#() -> c_13(h1#()):13

12:S:f2#() -> c_12(g2#())
-->_1 g2#() -> c_16(h2#()):16
-->_1 g2#() -> c_15(h1#()):15

13:S:g1#() -> c_13(h1#())
-->_1 h1#() -> c_17():17

14:S:g1#() -> c_14(h2#())
-->_1 h2#() -> c_18():18

15:S:g2#() -> c_15(h1#())
-->_1 h1#() -> c_17():17

16:S:g2#() -> c_16(h2#())
-->_1 h2#() -> c_18():18

17:S:h1#() -> c_17()

18:S:h2#() -> c_18()

The dependency graph contains no loops, we remove all dependency pairs.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0,e1#/5,e2#/5,e3#/11,e4#/11,e5#/4,f1#/0,f2#/0,g1#/0
,g2#/0,h1#/0,h2#/0} / {e6/3,i/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1
,c_13/1,c_14/1,c_15/1,c_16/1,c_17/0,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {e1#,e2#,e3#,e4#,e5#,f1#,f2#,g1#,g2#,h1#
,h2#} and constructors {e6,i}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))