```* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1} / {0/0,p/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,+,minus} and constructors {0,p,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
*#(0(),y) -> c_1()
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
+#(0(),y) -> c_4()
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
minus#(0()) -> c_7()
minus#(p(x)) -> c_8(minus#(x))
minus#(s(x)) -> c_9(minus#(x))
Weak DPs

and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
*#(0(),y) -> c_1()
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
+#(0(),y) -> c_4()
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
minus#(0()) -> c_7()
minus#(p(x)) -> c_8(minus#(x))
minus#(s(x)) -> c_9(minus#(x))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,4,7}
by application of
Pre({1,4,7}) = {2,3,5,6,8,9}.
Here rules are labelled as follows:
1: *#(0(),y) -> c_1()
2: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
3: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
4: +#(0(),y) -> c_4()
5: +#(p(x),y) -> c_5(+#(x,y))
6: +#(s(x),y) -> c_6(+#(x,y))
7: minus#(0()) -> c_7()
8: minus#(p(x)) -> c_8(minus#(x))
9: minus#(s(x)) -> c_9(minus#(x))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
minus#(p(x)) -> c_8(minus#(x))
minus#(s(x)) -> c_9(minus#(x))
- Weak DPs:
*#(0(),y) -> c_1()
+#(0(),y) -> c_4()
minus#(0()) -> c_7()
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
-->_3 minus#(s(x)) -> c_9(minus#(x)):6
-->_3 minus#(p(x)) -> c_8(minus#(x)):5
-->_1 +#(s(x),y) -> c_6(+#(x,y)):4
-->_1 +#(p(x),y) -> c_5(+#(x,y)):3
-->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
-->_3 minus#(0()) -> c_7():9
-->_1 +#(0(),y) -> c_4():8
-->_2 *#(0(),y) -> c_1():7
-->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):1

2:S:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):4
-->_1 +#(p(x),y) -> c_5(+#(x,y)):3
-->_1 +#(0(),y) -> c_4():8
-->_2 *#(0(),y) -> c_1():7
-->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
-->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):1

3:S:+#(p(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):4
-->_1 +#(0(),y) -> c_4():8
-->_1 +#(p(x),y) -> c_5(+#(x,y)):3

4:S:+#(s(x),y) -> c_6(+#(x,y))
-->_1 +#(0(),y) -> c_4():8
-->_1 +#(s(x),y) -> c_6(+#(x,y)):4
-->_1 +#(p(x),y) -> c_5(+#(x,y)):3

5:S:minus#(p(x)) -> c_8(minus#(x))
-->_1 minus#(s(x)) -> c_9(minus#(x)):6
-->_1 minus#(0()) -> c_7():9
-->_1 minus#(p(x)) -> c_8(minus#(x)):5

6:S:minus#(s(x)) -> c_9(minus#(x))
-->_1 minus#(0()) -> c_7():9
-->_1 minus#(s(x)) -> c_9(minus#(x)):6
-->_1 minus#(p(x)) -> c_8(minus#(x)):5

7:W:*#(0(),y) -> c_1()

8:W:+#(0(),y) -> c_4()

9:W:minus#(0()) -> c_7()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: *#(0(),y) -> c_1()
8: +#(0(),y) -> c_4()
9: minus#(0()) -> c_7()
* Step 4: PredecessorEstimationCP WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
minus#(p(x)) -> c_8(minus#(x))
minus#(s(x)) -> c_9(minus#(x))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
5: minus#(p(x)) -> c_8(minus#(x))

The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
minus#(p(x)) -> c_8(minus#(x))
minus#(s(x)) -> c_9(minus#(x))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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) = {1,2,3},
uargs(c_3) = {1,2},
uargs(c_5) = {1},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1}

Following symbols are considered usable:
{*#,+#,minus#}
TcT has computed the following interpretation:
p(*) = 1 + 2*x2^2
p(+) = 1 + x1 + x1^2 + x2
p(0) = 0
p(minus) = 1
p(p) = 1 + x1
p(s) = x1
p(*#) = 3*x1 + 2*x1*x2 + x1^2
p(+#) = 0
p(minus#) = 2*x1
p(c_1) = 0
p(c_2) = x1 + x2 + x3
p(c_3) = x1 + x2
p(c_4) = 0
p(c_5) = x1
p(c_6) = x1
p(c_7) = 0
p(c_8) = x1
p(c_9) = x1

Following rules are strictly oriented:
*#(p(x),y) = 4 + 5*x + 2*x*y + x^2 + 2*y
> 3*x + 2*x*y + x^2 + 2*y
= c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))

minus#(p(x)) = 2 + 2*x
> 2*x
= c_8(minus#(x))

Following rules are (at-least) weakly oriented:
*#(s(x),y) =  3*x + 2*x*y + x^2
>= 3*x + 2*x*y + x^2
=  c_3(+#(*(x,y),y),*#(x,y))

+#(p(x),y) =  0
>= 0
=  c_5(+#(x,y))

+#(s(x),y) =  0
>= 0
=  c_6(+#(x,y))

minus#(s(x)) =  2*x
>= 2*x
=  c_9(minus#(x))

** Step 4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
minus#(s(x)) -> c_9(minus#(x))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
minus#(p(x)) -> c_8(minus#(x))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
minus#(s(x)) -> c_9(minus#(x))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
minus#(p(x)) -> c_8(minus#(x))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
4: minus#(s(x)) -> c_9(minus#(x))

The strictly oriented rules are moved into the weak component.
*** Step 4.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
minus#(s(x)) -> c_9(minus#(x))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
minus#(p(x)) -> c_8(minus#(x))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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) = {1,2,3},
uargs(c_3) = {1,2},
uargs(c_5) = {1},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1}

Following symbols are considered usable:
{*#,+#,minus#}
TcT has computed the following interpretation:
p(*) = 1 + 2*x1
p(+) = x1*x2 + 3*x2 + 3*x2^2
p(0) = 0
p(minus) = 0
p(p) = 1 + x1
p(s) = 1 + x1
p(*#) = 2 + 2*x1 + x1*x2 + 2*x1^2 + 3*x2 + 2*x2^2
p(+#) = 0
p(minus#) = 2 + x1
p(c_1) = 1
p(c_2) = 1 + x1 + x2 + x3
p(c_3) = x1 + x2
p(c_4) = 0
p(c_5) = x1
p(c_6) = x1
p(c_7) = 0
p(c_8) = x1
p(c_9) = x1

Following rules are strictly oriented:
*#(s(x),y) = 6 + 6*x + x*y + 2*x^2 + 4*y + 2*y^2
> 2 + 2*x + x*y + 2*x^2 + 3*y + 2*y^2
= c_3(+#(*(x,y),y),*#(x,y))

minus#(s(x)) = 3 + x
> 2 + x
= c_9(minus#(x))

Following rules are (at-least) weakly oriented:
*#(p(x),y) =  6 + 6*x + x*y + 2*x^2 + 4*y + 2*y^2
>= 5 + 2*x + x*y + 2*x^2 + 4*y + 2*y^2
=  c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))

+#(p(x),y) =  0
>= 0
=  c_5(+#(x,y))

+#(s(x),y) =  0
>= 0
=  c_6(+#(x,y))

minus#(p(x)) =  3 + x
>= 2 + x
=  c_8(minus#(x))

*** Step 4.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
minus#(p(x)) -> c_8(minus#(x))
minus#(s(x)) -> c_9(minus#(x))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 4.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
minus#(p(x)) -> c_8(minus#(x))
minus#(s(x)) -> c_9(minus#(x))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:+#(p(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):2
-->_1 +#(p(x),y) -> c_5(+#(x,y)):1

2:S:+#(s(x),y) -> c_6(+#(x,y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):2
-->_1 +#(p(x),y) -> c_5(+#(x,y)):1

3:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
-->_3 minus#(s(x)) -> c_9(minus#(x)):6
-->_3 minus#(p(x)) -> c_8(minus#(x)):5
-->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
-->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
-->_1 +#(s(x),y) -> c_6(+#(x,y)):2
-->_1 +#(p(x),y) -> c_5(+#(x,y)):1

4:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
-->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
-->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
-->_1 +#(s(x),y) -> c_6(+#(x,y)):2
-->_1 +#(p(x),y) -> c_5(+#(x,y)):1

5:W:minus#(p(x)) -> c_8(minus#(x))
-->_1 minus#(s(x)) -> c_9(minus#(x)):6
-->_1 minus#(p(x)) -> c_8(minus#(x)):5

6:W:minus#(s(x)) -> c_9(minus#(x))
-->_1 minus#(s(x)) -> c_9(minus#(x)):6
-->_1 minus#(p(x)) -> c_8(minus#(x)):5

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: minus#(s(x)) -> c_9(minus#(x))
5: minus#(p(x)) -> c_8(minus#(x))
*** Step 4.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:+#(p(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):2
-->_1 +#(p(x),y) -> c_5(+#(x,y)):1

2:S:+#(s(x),y) -> c_6(+#(x,y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):2
-->_1 +#(p(x),y) -> c_5(+#(x,y)):1

3:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
-->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
-->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
-->_1 +#(s(x),y) -> c_6(+#(x,y)):2
-->_1 +#(p(x),y) -> c_5(+#(x,y)):1

4:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
-->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
-->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
-->_1 +#(s(x),y) -> c_6(+#(x,y)):2
-->_1 +#(p(x),y) -> c_5(+#(x,y)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
*** Step 4.b:1.b:3: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
and a lower component
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
Further, following extension rules are added to the lower component.
*#(p(x),y) -> *#(x,y)
*#(p(x),y) -> +#(*(x,y),minus(y))
*#(s(x),y) -> *#(x,y)
*#(s(x),y) -> +#(*(x,y),y)
**** Step 4.b:1.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))

The strictly oriented rules are moved into the weak component.
***** Step 4.b:1.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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_2) = {1,2},
uargs(c_3) = {1,2}

Following symbols are considered usable:
{*#,+#,minus#}
TcT has computed the following interpretation:
p(*) = [4] x1 + [0]
p(+) = [3] x1 + [2] x2 + [8]
p(0) = [1]
p(minus) = [8]
p(p) = [1] x1 + [1]
p(s) = [1] x1 + [0]
p(*#) = [8] x1 + [0]
p(+#) = [0]
p(minus#) = [0]
p(c_1) = [2]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [2] x1 + [1] x2 + [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]

Following rules are strictly oriented:
*#(p(x),y) = [8] x + [8]
> [8] x + [0]
= c_2(+#(*(x,y),minus(y)),*#(x,y))

Following rules are (at-least) weakly oriented:
*#(s(x),y) =  [8] x + [0]
>= [8] x + [0]
=  c_3(+#(*(x,y),y),*#(x,y))

***** Step 4.b:1.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 4.b:1.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))

The strictly oriented rules are moved into the weak component.
****** Step 4.b:1.b:3.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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_2) = {1,2},
uargs(c_3) = {1,2}

Following symbols are considered usable:
{*#,+#,minus#}
TcT has computed the following interpretation:
p(*) = [12] x1 + [2] x2 + [0]
p(+) = [2] x1 + [1]
p(0) = [0]
p(minus) = [10] x1 + [9]
p(p) = [1] x1 + [1]
p(s) = [1] x1 + [1]
p(*#) = [4] x1 + [2] x2 + [0]
p(+#) = [0]
p(minus#) = [0]
p(c_1) = [0]
p(c_2) = [4] x1 + [1] x2 + [2]
p(c_3) = [4] x1 + [1] x2 + [0]
p(c_4) = [8]
p(c_5) = [1] x1 + [0]
p(c_6) = [2] x1 + [1]
p(c_7) = [0]
p(c_8) = [2] x1 + [0]
p(c_9) = [1]

Following rules are strictly oriented:
*#(s(x),y) = [4] x + [2] y + [4]
> [4] x + [2] y + [0]
= c_3(+#(*(x,y),y),*#(x,y))

Following rules are (at-least) weakly oriented:
*#(p(x),y) =  [4] x + [2] y + [4]
>= [4] x + [2] y + [2]
=  c_2(+#(*(x,y),minus(y)),*#(x,y))

****** Step 4.b:1.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

****** Step 4.b:1.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
-->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
-->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)):1

2:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
-->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
-->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
2: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
****** Step 4.b:1.b:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

**** Step 4.b:1.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
- Weak DPs:
*#(p(x),y) -> *#(x,y)
*#(p(x),y) -> +#(*(x,y),minus(y))
*#(s(x),y) -> *#(x,y)
*#(s(x),y) -> +#(*(x,y),y)
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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:
2: +#(s(x),y) -> c_6(+#(x,y))

The strictly oriented rules are moved into the weak component.
***** Step 4.b:1.b:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
- Weak DPs:
*#(p(x),y) -> *#(x,y)
*#(p(x),y) -> +#(*(x,y),minus(y))
*#(s(x),y) -> *#(x,y)
*#(s(x),y) -> +#(*(x,y),y)
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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_5) = {1},
uargs(c_6) = {1}

Following symbols are considered usable:
{*,+,minus,*#,+#,minus#}
TcT has computed the following interpretation:
p(*) = 2*x1*x2
p(+) = x1 + 2*x2
p(0) = 0
p(minus) = x1
p(p) = 1 + x1
p(s) = 1 + x1
p(*#) = 3 + 2*x1*x2
p(+#) = 3 + x1
p(minus#) = 2 + x1 + 2*x1^2
p(c_1) = 1
p(c_2) = x2
p(c_3) = 1
p(c_4) = 0
p(c_5) = 1 + x1
p(c_6) = x1
p(c_7) = 0
p(c_8) = 1 + x1
p(c_9) = 1

Following rules are strictly oriented:
+#(s(x),y) = 4 + x
> 3 + x
= c_6(+#(x,y))

Following rules are (at-least) weakly oriented:
*#(p(x),y) =  3 + 2*x*y + 2*y
>= 3 + 2*x*y
=  *#(x,y)

*#(p(x),y) =  3 + 2*x*y + 2*y
>= 3 + 2*x*y
=  +#(*(x,y),minus(y))

*#(s(x),y) =  3 + 2*x*y + 2*y
>= 3 + 2*x*y
=  *#(x,y)

*#(s(x),y) =  3 + 2*x*y + 2*y
>= 3 + 2*x*y
=  +#(*(x,y),y)

+#(p(x),y) =  4 + x
>= 4 + x
=  c_5(+#(x,y))

*(0(),y) =  0
>= 0
=  0()

*(p(x),y) =  2*x*y + 2*y
>= 2*x*y + 2*y
=  +(*(x,y),minus(y))

*(s(x),y) =  2*x*y + 2*y
>= 2*x*y + 2*y
=  +(*(x,y),y)

+(0(),y) =  2*y
>= y
=  y

+(p(x),y) =  1 + x + 2*y
>= 1 + x + 2*y
=  p(+(x,y))

+(s(x),y) =  1 + x + 2*y
>= 1 + x + 2*y
=  s(+(x,y))

minus(0()) =  0
>= 0
=  0()

minus(p(x)) =  1 + x
>= 1 + x
=  s(minus(x))

minus(s(x)) =  1 + x
>= 1 + x
=  p(minus(x))

***** Step 4.b:1.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
- Weak DPs:
*#(p(x),y) -> *#(x,y)
*#(p(x),y) -> +#(*(x,y),minus(y))
*#(s(x),y) -> *#(x,y)
*#(s(x),y) -> +#(*(x,y),y)
+#(s(x),y) -> c_6(+#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 4.b:1.b:3.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
- Weak DPs:
*#(p(x),y) -> *#(x,y)
*#(p(x),y) -> +#(*(x,y),minus(y))
*#(s(x),y) -> *#(x,y)
*#(s(x),y) -> +#(*(x,y),y)
+#(s(x),y) -> c_6(+#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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: +#(p(x),y) -> c_5(+#(x,y))

The strictly oriented rules are moved into the weak component.
****** Step 4.b:1.b:3.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
+#(p(x),y) -> c_5(+#(x,y))
- Weak DPs:
*#(p(x),y) -> *#(x,y)
*#(p(x),y) -> +#(*(x,y),minus(y))
*#(s(x),y) -> *#(x,y)
*#(s(x),y) -> +#(*(x,y),y)
+#(s(x),y) -> c_6(+#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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_5) = {1},
uargs(c_6) = {1}

Following symbols are considered usable:
{*,+,minus,*#,+#,minus#}
TcT has computed the following interpretation:
p(*) = x1*x2 + 2*x1^2
p(+) = 1 + x1 + x2
p(0) = 0
p(minus) = x1
p(p) = 1 + x1
p(s) = 1 + x1
p(*#) = x1*x2 + 2*x1^2 + 3*x2 + 3*x2^2
p(+#) = 2 + x1 + x2 + 2*x2^2
p(minus#) = 2*x1 + x1^2
p(c_1) = 0
p(c_2) = x1
p(c_3) = 0
p(c_4) = 0
p(c_5) = x1
p(c_6) = x1
p(c_7) = 0
p(c_8) = 0
p(c_9) = 1

Following rules are strictly oriented:
+#(p(x),y) = 3 + x + y + 2*y^2
> 2 + x + y + 2*y^2
= c_5(+#(x,y))

Following rules are (at-least) weakly oriented:
*#(p(x),y) =  2 + 4*x + x*y + 2*x^2 + 4*y + 3*y^2
>= x*y + 2*x^2 + 3*y + 3*y^2
=  *#(x,y)

*#(p(x),y) =  2 + 4*x + x*y + 2*x^2 + 4*y + 3*y^2
>= 2 + x*y + 2*x^2 + y + 2*y^2
=  +#(*(x,y),minus(y))

*#(s(x),y) =  2 + 4*x + x*y + 2*x^2 + 4*y + 3*y^2
>= x*y + 2*x^2 + 3*y + 3*y^2
=  *#(x,y)

*#(s(x),y) =  2 + 4*x + x*y + 2*x^2 + 4*y + 3*y^2
>= 2 + x*y + 2*x^2 + y + 2*y^2
=  +#(*(x,y),y)

+#(s(x),y) =  3 + x + y + 2*y^2
>= 2 + x + y + 2*y^2
=  c_6(+#(x,y))

*(0(),y) =  0
>= 0
=  0()

*(p(x),y) =  2 + 4*x + x*y + 2*x^2 + y
>= 1 + x*y + 2*x^2 + y
=  +(*(x,y),minus(y))

*(s(x),y) =  2 + 4*x + x*y + 2*x^2 + y
>= 1 + x*y + 2*x^2 + y
=  +(*(x,y),y)

+(0(),y) =  1 + y
>= y
=  y

+(p(x),y) =  2 + x + y
>= 2 + x + y
=  p(+(x,y))

+(s(x),y) =  2 + x + y
>= 2 + x + y
=  s(+(x,y))

minus(0()) =  0
>= 0
=  0()

minus(p(x)) =  1 + x
>= 1 + x
=  s(minus(x))

minus(s(x)) =  1 + x
>= 1 + x
=  p(minus(x))

****** Step 4.b:1.b:3.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
*#(p(x),y) -> *#(x,y)
*#(p(x),y) -> +#(*(x,y),minus(y))
*#(s(x),y) -> *#(x,y)
*#(s(x),y) -> +#(*(x,y),y)
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

****** Step 4.b:1.b:3.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
*#(p(x),y) -> *#(x,y)
*#(p(x),y) -> +#(*(x,y),minus(y))
*#(s(x),y) -> *#(x,y)
*#(s(x),y) -> +#(*(x,y),y)
+#(p(x),y) -> c_5(+#(x,y))
+#(s(x),y) -> c_6(+#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:*#(p(x),y) -> *#(x,y)
-->_1 *#(s(x),y) -> +#(*(x,y),y):4
-->_1 *#(s(x),y) -> *#(x,y):3
-->_1 *#(p(x),y) -> +#(*(x,y),minus(y)):2
-->_1 *#(p(x),y) -> *#(x,y):1

2:W:*#(p(x),y) -> +#(*(x,y),minus(y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):6
-->_1 +#(p(x),y) -> c_5(+#(x,y)):5

3:W:*#(s(x),y) -> *#(x,y)
-->_1 *#(s(x),y) -> +#(*(x,y),y):4
-->_1 *#(s(x),y) -> *#(x,y):3
-->_1 *#(p(x),y) -> +#(*(x,y),minus(y)):2
-->_1 *#(p(x),y) -> *#(x,y):1

4:W:*#(s(x),y) -> +#(*(x,y),y)
-->_1 +#(s(x),y) -> c_6(+#(x,y)):6
-->_1 +#(p(x),y) -> c_5(+#(x,y)):5

5:W:+#(p(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):6
-->_1 +#(p(x),y) -> c_5(+#(x,y)):5

6:W:+#(s(x),y) -> c_6(+#(x,y))
-->_1 +#(s(x),y) -> c_6(+#(x,y)):6
-->_1 +#(p(x),y) -> c_5(+#(x,y)):5

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: *#(p(x),y) -> *#(x,y)
3: *#(s(x),y) -> *#(x,y)
2: *#(p(x),y) -> +#(*(x,y),minus(y))
4: *#(s(x),y) -> +#(*(x,y),y)
6: +#(s(x),y) -> c_6(+#(x,y))
5: +#(p(x),y) -> c_5(+#(x,y))
****** Step 4.b:1.b:3.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
*(0(),y) -> 0()
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0(),y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0()) -> 0()
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
- Signature:
{*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))
```