```* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(X))) -> s(half(X))
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)))
- Signature:
- Obligation:
innermost runtime complexity wrt. defined symbols {add,dbl,first,half,sqr,terms} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(s(0())) -> c_8()
half#(s(s(X))) -> c_9(half#(X))
sqr#(0()) -> c_10()
terms#(N) -> c_12(sqr#(N))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(s(0())) -> c_8()
half#(s(s(X))) -> c_9(half#(X))
sqr#(0()) -> c_10()
terms#(N) -> c_12(sqr#(N))
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(X))) -> s(half(X))
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(s(0())) -> c_8()
half#(s(s(X))) -> c_9(half#(X))
sqr#(0()) -> c_10()
terms#(N) -> c_12(sqr#(N))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(s(0())) -> c_8()
half#(s(s(X))) -> c_9(half#(X))
sqr#(0()) -> c_10()
terms#(N) -> c_12(sqr#(N))
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,5,6,7,8,10}
by application of
Pre({1,3,5,6,7,8,10}) = {2,4,9,11,12}.
Here rules are labelled as follows:
3: dbl#(0()) -> c_3()
4: dbl#(s(X)) -> c_4(dbl#(X))
5: first#(0(),X) -> c_5()
6: first#(s(X),cons(Y)) -> c_6()
7: half#(0()) -> c_7()
8: half#(s(0())) -> c_8()
9: half#(s(s(X))) -> c_9(half#(X))
10: sqr#(0()) -> c_10()
12: terms#(N) -> c_12(sqr#(N))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
terms#(N) -> c_12(sqr#(N))
- Weak DPs:
dbl#(0()) -> c_3()
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(s(0())) -> c_8()
sqr#(0()) -> c_10()
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph

2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(0()) -> c_3():7
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2

3:S:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(0())) -> c_8():11
-->_1 half#(0()) -> c_7():10
-->_1 half#(s(s(X))) -> c_9(half#(X)):3

-->_2 sqr#(0()) -> c_10():12
-->_3 dbl#(0()) -> c_3():7
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2

5:S:terms#(N) -> c_12(sqr#(N))
-->_1 sqr#(0()) -> c_10():12

7:W:dbl#(0()) -> c_3()

8:W:first#(0(),X) -> c_5()

9:W:first#(s(X),cons(Y)) -> c_6()

10:W:half#(0()) -> c_7()

11:W:half#(s(0())) -> c_8()

12:W:sqr#(0()) -> c_10()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: first#(s(X),cons(Y)) -> c_6()
8: first#(0(),X) -> c_5()
12: sqr#(0()) -> c_10()
10: half#(0()) -> c_7()
11: half#(s(0())) -> c_8()
7: dbl#(0()) -> c_3()
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
terms#(N) -> c_12(sqr#(N))
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
+ Details:
Consider the dependency graph

2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2

3:S:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):3

-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2

5:S:terms#(N) -> c_12(sqr#(N))

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).

[(5,terms#(N) -> c_12(sqr#(N)))]
* Step 6: Decompose WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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:
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}

Problem (S)
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
- Weak DPs:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph

2:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2

3:W:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):3

-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: half#(s(s(X))) -> c_9(half#(X))
2: dbl#(s(X)) -> c_4(dbl#(X))
** Step 6.a:2: SimplifyRHS WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
- Weak DPs:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
** Step 6.a:3: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
- Weak DPs:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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
and a lower component
Further, following extension rules are added to the lower component.
sqr#(s(X)) -> sqr#(X)
*** Step 6.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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:

The strictly oriented rules are moved into the weak component.
**** Step 6.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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_11) = {1,2}

Following symbols are considered usable:
TcT has computed the following interpretation:
p(0) = [1]
p(cons) = [1] x1 + [0]
p(dbl) = [8] x1 + [5]
p(first) = [2] x1 + [1] x2 + [0]
p(half) = [0]
p(nil) = [0]
p(recip) = [0]
p(s) = [1] x1 + [2]
p(sqr) = [8] x1 + [0]
p(terms) = [0]
p(dbl#) = [1] x1 + [0]
p(first#) = [1] x1 + [0]
p(half#) = [1] x1 + [0]
p(sqr#) = [4] x1 + [0]
p(terms#) = [1] x1 + [0]
p(c_1) = [0]
p(c_2) = [8] x1 + [0]
p(c_3) = [0]
p(c_4) = [2] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [4] x1 + [1] x2 + [2]
p(c_12) = [0]

Following rules are strictly oriented:
sqr#(s(X)) = [4] X + [8]
> [4] X + [6]

Following rules are (at-least) weakly oriented:
dbl(0()) =  [13]
>= [1]
=  0()

dbl(s(X)) =  [8] X + [21]
>= [8] X + [9]
=  s(s(dbl(X)))

**** Step 6.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 6.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
**** Step 6.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

*** Step 6.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
- Weak DPs:
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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:

The strictly oriented rules are moved into the weak component.
**** Step 6.a:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
- Weak DPs:
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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_2) = {1}

Following symbols are considered usable:
TcT has computed the following interpretation:
p(0) = 0
p(add) = 1 + x1 + 3*x2
p(cons) = x1
p(dbl) = 2*x1
p(first) = 1 + x1 + 2*x1*x2 + x1^2 + 4*x2^2
p(half) = 1
p(nil) = 1
p(recip) = 0
p(s) = 1 + x1
p(sqr) = 3*x1^2
p(terms) = 4 + x1 + 4*x1^2
p(add#) = 2 + 2*x1 + 2*x2
p(dbl#) = x1
p(first#) = x1 + x1*x2 + 4*x2 + x2^2
p(half#) = 1 + 4*x1
p(sqr#) = 4 + 2*x1 + 6*x1^2
p(terms#) = x1 + 2*x1^2
p(c_1) = 1
p(c_2) = x1
p(c_3) = 1
p(c_4) = 1
p(c_5) = 1
p(c_6) = 0
p(c_7) = 0
p(c_8) = 0
p(c_9) = x1
p(c_10) = 0
p(c_11) = 1 + x1
p(c_12) = 1

Following rules are strictly oriented:
add#(s(X),Y) = 4 + 2*X + 2*Y
> 2 + 2*X + 2*Y

Following rules are (at-least) weakly oriented:
sqr#(s(X)) =  12 + 14*X + 6*X^2
>= 2 + 4*X + 6*X^2

sqr#(s(X)) =  12 + 14*X + 6*X^2
>= 4 + 2*X + 6*X^2
=  sqr#(X)

>= X
=  X

add(s(X),Y) =  2 + X + 3*Y
>= 2 + X + 3*Y

dbl(0()) =  0
>= 0
=  0()

dbl(s(X)) =  2 + 2*X
>= 2 + 2*X
=  s(s(dbl(X)))

sqr(0()) =  0
>= 0
=  0()

sqr(s(X)) =  3 + 6*X + 3*X^2
>= 2 + 6*X + 3*X^2

**** Step 6.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 6.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph

3:W:sqr#(s(X)) -> sqr#(X)
-->_1 sqr#(s(X)) -> sqr#(X):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: sqr#(s(X)) -> sqr#(X)
**** Step 6.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
- Weak DPs:
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1

2:S:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):2

-->_3 dbl#(s(X)) -> c_4(dbl#(X)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/3,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1

2:S:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):2

-->_3 dbl#(s(X)) -> c_4(dbl#(X)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
** Step 6.b:3: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Weak TRS:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
** Step 6.b:4: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}

Problem (S)
- Strict DPs:
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
*** Step 6.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1

2:W:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):2

3:W:sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):1
-->_1 sqr#(s(X)) -> c_11(sqr#(X),dbl#(X)):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: half#(s(s(X))) -> c_9(half#(X))
*** Step 6.b:4.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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: dbl#(s(X)) -> c_4(dbl#(X))

The strictly oriented rules are moved into the weak component.
**** Step 6.b:4.a:2.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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_4) = {1},
uargs(c_11) = {1,2}

Following symbols are considered usable:
TcT has computed the following interpretation:
p(0) = 1
p(add) = x1*x2 + x1^2 + 2*x2^2
p(cons) = x1
p(dbl) = x1 + x1^2
p(first) = 1 + x1 + x2 + x2^2
p(half) = 8 + 2*x1 + x1^2
p(nil) = 0
p(recip) = 1
p(s) = 1 + x1
p(sqr) = 4 + 8*x1^2
p(terms) = x1
p(add#) = 1 + 2*x1 + 8*x1*x2
p(dbl#) = 8 + 2*x1
p(first#) = 1 + x1 + x1^2 + 2*x2 + x2^2
p(half#) = 1 + x1
p(sqr#) = 5 + x1 + 12*x1^2
p(terms#) = 0
p(c_1) = 0
p(c_2) = 1
p(c_3) = 1
p(c_4) = x1
p(c_5) = 1
p(c_6) = 0
p(c_7) = 0
p(c_8) = 0
p(c_9) = 0
p(c_10) = 0
p(c_11) = x1 + x2
p(c_12) = 1

Following rules are strictly oriented:
dbl#(s(X)) = 10 + 2*X
> 8 + 2*X
= c_4(dbl#(X))

Following rules are (at-least) weakly oriented:
sqr#(s(X)) =  18 + 25*X + 12*X^2
>= 13 + 3*X + 12*X^2
=  c_11(sqr#(X),dbl#(X))

**** Step 6.b:4.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 6.b:4.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1

2:W:sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
-->_1 sqr#(s(X)) -> c_11(sqr#(X),dbl#(X)):2
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
1: dbl#(s(X)) -> c_4(dbl#(X))
**** Step 6.b:4.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

*** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):1

2:S:sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):3
-->_1 sqr#(s(X)) -> c_11(sqr#(X),dbl#(X)):2

3:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: dbl#(s(X)) -> c_4(dbl#(X))
*** Step 6.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):1

2:S:sqr#(s(X)) -> c_11(sqr#(X),dbl#(X))
-->_1 sqr#(s(X)) -> c_11(sqr#(X),dbl#(X)):2

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_11(sqr#(X))
*** Step 6.b:4.b:3: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_9(half#(X))
sqr#(s(X)) -> c_11(sqr#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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:
half#(s(s(X))) -> c_9(half#(X))
- Weak DPs:
sqr#(s(X)) -> c_11(sqr#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}

Problem (S)
- Strict DPs:
sqr#(s(X)) -> c_11(sqr#(X))
- Weak DPs:
half#(s(s(X))) -> c_9(half#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
**** Step 6.b:4.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_9(half#(X))
- Weak DPs:
sqr#(s(X)) -> c_11(sqr#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):1

2:W:sqr#(s(X)) -> c_11(sqr#(X))
-->_1 sqr#(s(X)) -> c_11(sqr#(X)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: sqr#(s(X)) -> c_11(sqr#(X))
**** Step 6.b:4.b:3.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_9(half#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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: half#(s(s(X))) -> c_9(half#(X))

The strictly oriented rules are moved into the weak component.
***** Step 6.b:4.b:3.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_9(half#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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_9) = {1}

Following symbols are considered usable:
TcT has computed the following interpretation:
p(0) = [2]
p(add) = [8] x1 + [1]
p(cons) = [1] x1 + [0]
p(dbl) = [2] x1 + [0]
p(first) = [1] x1 + [0]
p(half) = [4]
p(nil) = [1]
p(recip) = [0]
p(s) = [1] x1 + [1]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [0]
p(add#) = [1] x1 + [0]
p(dbl#) = [1] x1 + [2]
p(first#) = [4] x1 + [1] x2 + [0]
p(half#) = [4] x1 + [1]
p(sqr#) = [0]
p(terms#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [4] x1 + [0]
p(c_3) = [1]
p(c_4) = [2]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [2]
p(c_8) = [1]
p(c_9) = [1] x1 + [6]
p(c_10) = [0]
p(c_11) = [8] x1 + [1]
p(c_12) = [1]

Following rules are strictly oriented:
half#(s(s(X))) = [4] X + [9]
> [4] X + [7]
= c_9(half#(X))

Following rules are (at-least) weakly oriented:

***** Step 6.b:4.b:3.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(X))) -> c_9(half#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 6.b:4.b:3.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(X))) -> c_9(half#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: half#(s(s(X))) -> c_9(half#(X))
***** Step 6.b:4.b:3.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

**** Step 6.b:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_11(sqr#(X))
- Weak DPs:
half#(s(s(X))) -> c_9(half#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:sqr#(s(X)) -> c_11(sqr#(X))
-->_1 sqr#(s(X)) -> c_11(sqr#(X)):1

2:W:half#(s(s(X))) -> c_9(half#(X))
-->_1 half#(s(s(X))) -> c_9(half#(X)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: half#(s(s(X))) -> c_9(half#(X))
**** Step 6.b:4.b:3.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_11(sqr#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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: sqr#(s(X)) -> c_11(sqr#(X))

The strictly oriented rules are moved into the weak component.
***** Step 6.b:4.b:3.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_11(sqr#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,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_11) = {1}

Following symbols are considered usable:
TcT has computed the following interpretation:
p(0) = [0]
p(cons) = [1] x1 + [0]
p(dbl) = [8] x1 + [0]
p(first) = [4] x2 + [1]
p(half) = [1]
p(nil) = [1]
p(recip) = [1]
p(s) = [1] x1 + [7]
p(sqr) = [1] x1 + [0]
p(terms) = [1]
p(add#) = [1] x1 + [8]
p(dbl#) = [2]
p(first#) = [0]
p(half#) = [0]
p(sqr#) = [4] x1 + [1]
p(terms#) = [2] x1 + [1]
p(c_1) = [0]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [1] x1 + [0]
p(c_5) = [1]
p(c_6) = [0]
p(c_7) = [4]
p(c_8) = [0]
p(c_9) = [4] x1 + [1]
p(c_10) = [8]
p(c_11) = [1] x1 + [14]
p(c_12) = [2] x1 + [4]

Following rules are strictly oriented:
sqr#(s(X)) = [4] X + [29]
> [4] X + [15]
= c_11(sqr#(X))

Following rules are (at-least) weakly oriented:

***** Step 6.b:4.b:3.b:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> c_11(sqr#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 6.b:4.b:3.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> c_11(sqr#(X))
- Signature:
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:sqr#(s(X)) -> c_11(sqr#(X))
-->_1 sqr#(s(X)) -> c_11(sqr#(X)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sqr#(s(X)) -> c_11(sqr#(X))
***** Step 6.b:4.b:3.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature: