MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
f(0(),0(),0(),0(),0()) -> 0()
f(0(),0(),0(),0(),s(x5)) -> f(x5,x5,x5,x5,x5)
f(0(),0(),0(),s(x4),x5) -> f(x4,x4,x4,x4,x5)
f(0(),0(),s(x3),x4,x5) -> f(x3,x3,x3,x4,x5)
f(0(),s(x2),x3,x4,x5) -> f(x2,x2,x3,x4,x5)
f(s(x1),x2,x3,x4,x5) -> f(x1,x2,x3,x4,x5)
- Signature:
{f/5} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
f#(0(),0(),0(),0(),0()) -> c_1()
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
f#(0(),0(),0(),0(),0()) -> c_1()
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Strict TRS:
f(0(),0(),0(),0(),0()) -> 0()
f(0(),0(),0(),0(),s(x5)) -> f(x5,x5,x5,x5,x5)
f(0(),0(),0(),s(x4),x5) -> f(x4,x4,x4,x4,x5)
f(0(),0(),s(x3),x4,x5) -> f(x3,x3,x3,x4,x5)
f(0(),s(x2),x3,x4,x5) -> f(x2,x2,x3,x4,x5)
f(s(x1),x2,x3,x4,x5) -> f(x1,x2,x3,x4,x5)
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(0(),0(),0(),0(),0()) -> c_1()
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
f#(0(),0(),0(),0(),0()) -> c_1()
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {2,3,4,5,6}.
Here rules are labelled as follows:
1: f#(0(),0(),0(),0(),0()) -> c_1()
2: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
3: f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
4: f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
5: f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
6: f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Weak DPs:
f#(0(),0(),0(),0(),0()) -> c_1()
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
-->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5
-->_1 f#(0(),0(),0(),0(),0()) -> c_1():6
2:S:f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
-->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5
-->_1 f#(0(),0(),0(),0(),0()) -> c_1():6
-->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1
3:S:f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
-->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5
-->_1 f#(0(),0(),0(),0(),0()) -> c_1():6
-->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2
-->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1
4:S:f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
-->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5
-->_1 f#(0(),0(),0(),0(),0()) -> c_1():6
-->_1 f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)):3
-->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2
-->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1
5:S:f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
-->_1 f#(0(),0(),0(),0(),0()) -> c_1():6
-->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5
-->_1 f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)):4
-->_1 f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)):3
-->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2
-->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1
6:W:f#(0(),0(),0(),0(),0()) -> c_1()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: f#(0(),0(),0(),0(),0()) -> c_1()
* Step 5: PredecessorEstimationCP MAYBE
+ Considered Problem:
- Strict DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,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: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,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},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
{f#}
TcT has computed the following interpretation:
p(0) = [0]
p(f) = [1] x1 + [1] x2 + [1] x4 + [1] x5 + [8]
p(s) = [1] x1 + [4]
p(f#) = [2] x5 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1] x1 + [0]
Following rules are strictly oriented:
f#(0(),0(),0(),0(),s(x5)) = [2] x5 + [8]
> [2] x5 + [0]
= c_2(f#(x5,x5,x5,x5,x5))
Following rules are (at-least) weakly oriented:
f#(0(),0(),0(),s(x4),x5) = [2] x5 + [0]
>= [2] x5 + [0]
= c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) = [2] x5 + [0]
>= [2] x5 + [0]
= c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) = [2] x5 + [0]
>= [2] x5 + [0]
= c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) = [2] x5 + [0]
>= [2] x5 + [0]
= c_6(f#(x1,x2,x3,x4,x5))
** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Weak DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 5.b:1: PredecessorEstimationCP MAYBE
+ Considered Problem:
- Strict DPs:
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Weak DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,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: f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
The strictly oriented rules are moved into the weak component.
*** Step 5.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Weak DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,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},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
{f#}
TcT has computed the following interpretation:
p(0) = 0
p(f) = 4 + 2*x1*x3 + x2 + x2*x4 + 2*x2*x5 + 2*x2^2 + x3*x5 + 2*x3^2 + x4*x5 + x5
p(s) = 2 + x1
p(f#) = 6*x4 + 3*x5^2
p(c_1) = 1
p(c_2) = x1
p(c_3) = 4 + x1
p(c_4) = x1
p(c_5) = x1
p(c_6) = x1
Following rules are strictly oriented:
f#(0(),0(),0(),s(x4),x5) = 12 + 6*x4 + 3*x5^2
> 4 + 6*x4 + 3*x5^2
= c_3(f#(x4,x4,x4,x4,x5))
Following rules are (at-least) weakly oriented:
f#(0(),0(),0(),0(),s(x5)) = 12 + 12*x5 + 3*x5^2
>= 6*x5 + 3*x5^2
= c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),s(x3),x4,x5) = 6*x4 + 3*x5^2
>= 6*x4 + 3*x5^2
= c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) = 6*x4 + 3*x5^2
>= 6*x4 + 3*x5^2
= c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) = 6*x4 + 3*x5^2
>= 6*x4 + 3*x5^2
= c_6(f#(x1,x2,x3,x4,x5))
*** Step 5.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Weak DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 5.b:1.b:1: NaturalMI MAYBE
+ Considered Problem:
- Strict DPs:
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Weak DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 4, miDegree = 4, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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},
uargs(c_5) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
{f#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
[0]
p(f) = [0]
[0]
[0]
[0]
p(s) = [1 1 1 1] [0]
[0 0 0 0] x1 + [0]
[0 0 1 1] [0]
[0 0 0 1] [1]
p(f#) = [0 0 0 0] [0 0 0 0] [0 0 0 1] [0 0 1 1] [1 0 1 0] [0]
[0 0 0 0] x1 + [0 1 0 1] x2 + [0 0 1 1] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0]
[0 1 0 0] [1 1 0 0] [0 0 1 1] [1 0 0 0] [0 0 0 0] [0]
p(c_1) = [0]
[0]
[0]
[0]
p(c_2) = [1 0 1 0] [0]
[0 0 0 0] x1 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
p(c_3) = [1 0 0 0] [0]
[0 0 0 0] x1 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
p(c_4) = [1 0 0 0] [0]
[0 0 0 0] x1 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
p(c_5) = [1 0 0 0] [0]
[0 0 0 0] x1 + [1]
[0 0 1 0] [0]
[0 1 0 0] [0]
p(c_6) = [1 0 0 0] [0]
[0 0 0 0] x1 + [0]
[0 0 1 0] [0]
[0 0 0 0] [0]
Following rules are strictly oriented:
f#(0(),0(),s(x3),x4,x5) = [0 0 0 1] [0 0 1 1] [1 0 1 0] [1]
[0 0 1 2] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [1]
[0 0 0 0] [0 0 0 0] [0 1 0 0] [0]
[0 0 1 2] [1 0 0 0] [0 0 0 0] [1]
> [0 0 0 1] [0 0 1 1] [1 0 1 0] [0]
[0 0 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0]
= c_4(f#(x3,x3,x3,x4,x5))
Following rules are (at-least) weakly oriented:
f#(0(),0(),0(),0(),s(x5)) = [1 1 2 2] [0]
[0 0 0 0] x5 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
>= [1 1 2 2] [0]
[0 0 0 0] x5 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
= c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) = [0 0 1 2] [1 0 1 0] [1]
[0 0 0 0] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 1 0 0] [0]
[1 1 1 1] [0 0 0 0] [0]
>= [0 0 1 2] [1 0 1 0] [0]
[0 0 0 0] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 0 0 0] [0]
[0 0 0 0] [0 0 0 0] [0]
= c_3(f#(x4,x4,x4,x4,x5))
f#(0(),s(x2),x3,x4,x5) = [0 0 0 0] [0 0 0 1] [0 0 1 1] [1 0 1 0] [0]
[0 0 0 1] x2 + [0 0 1 1] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [1]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0]
[1 1 1 1] [0 0 1 1] [1 0 0 0] [0 0 0 0] [0]
>= [0 0 0 0] [0 0 0 1] [0 0 1 1] [1 0 1 0] [0]
[0 0 0 0] x2 + [0 0 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [1]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0]
[0 1 0 1] [0 0 1 1] [0 0 0 0] [0 0 0 0] [0]
= c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) = [0 0 0 0] [0 0 0 1] [0 0 1 1] [1 0 1 0] [0]
[0 1 0 1] x2 + [0 0 1 1] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0]
[1 1 0 0] [0 0 1 1] [1 0 0 0] [0 0 0 0] [0]
>= [0 0 0 1] [0 0 1 1] [1 0 1 0] [0]
[0 0 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 0 0 0] [0 1 0 0] [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0]
= c_6(f#(x1,x2,x3,x4,x5))
*** Step 5.b:1.b:2: NaturalMI MAYBE
+ Considered Problem:
- Strict DPs:
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Weak DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 4, miDegree = 4, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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},
uargs(c_5) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
{f#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[1]
[0]
p(f) = [0]
[0]
[0]
[0]
p(s) = [1 1 1 1] [0]
[0 1 1 1] x1 + [0]
[0 0 1 1] [1]
[0 0 0 1] [1]
p(f#) = [0 0 0 0] [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [0]
[0 1 0 0] x1 + [0 1 0 0] x2 + [0 1 0 0] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0]
[1 1 0 0] [0 0 1 0] [0 0 0 1] [0 0 0 0] [0 0 1 0] [1]
[0 0 0 0] [0 1 1 1] [0 1 0 0] [0 0 0 0] [0 0 0 0] [0]
p(c_1) = [0]
[0]
[0]
[0]
p(c_2) = [1 0 0 0] [1]
[0 0 0 0] x1 + [1]
[0 0 0 0] [1]
[0 0 0 0] [0]
p(c_3) = [1 0 0 0] [0]
[0 0 0 0] x1 + [1]
[0 0 0 0] [1]
[0 0 0 0] [1]
p(c_4) = [1 0 0 0] [1]
[0 0 0 0] x1 + [0]
[0 0 0 0] [1]
[0 0 0 0] [1]
p(c_5) = [1 0 0 0] [0]
[0 0 0 1] x1 + [0]
[0 0 0 0] [0]
[0 0 0 0] [1]
p(c_6) = [1 0 0 0] [0]
[0 1 0 0] x1 + [0]
[0 0 0 0] [1]
[0 0 0 0] [0]
Following rules are strictly oriented:
f#(0(),s(x2),x3,x4,x5) = [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [1]
[0 1 1 1] x2 + [0 1 0 0] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0]
[0 0 1 1] [0 0 0 1] [0 0 0 0] [0 0 1 0] [2]
[0 1 2 3] [0 1 0 0] [0 0 0 0] [0 0 0 0] [2]
> [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [0]
[0 1 1 1] x2 + [0 1 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1]
= c_5(f#(x2,x2,x3,x4,x5))
Following rules are (at-least) weakly oriented:
f#(0(),0(),0(),0(),s(x5)) = [1 1 2 3] [3]
[0 0 0 0] x5 + [1]
[0 0 1 1] [3]
[0 0 0 0] [1]
>= [1 1 2 2] [1]
[0 0 0 0] x5 + [1]
[0 0 0 0] [1]
[0 0 0 0] [0]
= c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) = [0 1 1 1] [1 0 1 1] [1]
[0 0 1 2] x4 + [0 0 0 0] x5 + [2]
[0 0 0 0] [0 0 1 0] [2]
[0 0 0 0] [0 0 0 0] [1]
>= [0 1 1 1] [1 0 1 1] [0]
[0 0 0 0] x4 + [0 0 0 0] x5 + [1]
[0 0 0 0] [0 0 0 0] [1]
[0 0 0 0] [0 0 0 0] [1]
= c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) = [0 0 1 1] [0 1 0 0] [1 0 1 1] [1]
[0 1 1 1] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0]
[0 0 0 1] [0 0 0 0] [0 0 1 0] [3]
[0 1 1 1] [0 0 0 0] [0 0 0 0] [1]
>= [0 0 1 1] [0 1 0 0] [1 0 1 1] [1]
[0 0 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [1]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [1]
= c_4(f#(x3,x3,x3,x4,x5))
f#(s(x1),x2,x3,x4,x5) = [0 0 0 0] [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [0]
[0 1 1 1] x1 + [0 1 0 0] x2 + [0 1 0 0] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0]
[1 2 2 2] [0 0 1 0] [0 0 0 1] [0 0 0 0] [0 0 1 0] [1]
[0 0 0 0] [0 1 1 1] [0 1 0 0] [0 0 0 0] [0 0 0 0] [0]
>= [0 0 0 0] [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [0]
[0 1 0 0] x1 + [0 1 0 0] x2 + [0 1 0 0] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0]
= c_6(f#(x1,x2,x3,x4,x5))
*** Step 5.b:1.b:3: Failure MAYBE
+ Considered Problem:
- Strict DPs:
f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5))
- Weak DPs:
f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5))
f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5))
f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5))
f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5))
- Signature:
{f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE