MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
help(x,y) -> ifb(lt(y,x),x,y)
ifa(false(),x) -> logZeroError()
ifa(true(),x) -> help(x,1())
ifb(false(),x,y) -> y
ifb(true(),x,y) -> help(half(x),s(y))
logarithm(x) -> ifa(lt(0(),x),x)
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{half/1,help/2,ifa/2,ifb/3,logarithm/1,lt/2} / {0/0,1/0,false/0,logZeroError/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,help,ifa,ifb,logarithm,lt} and constructors {0,1
,false,logZeroError,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
half#(0()) -> c_1()
half#(s(0())) -> c_2()
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(false(),x) -> c_5()
ifa#(true(),x) -> c_6(help#(x,1()))
ifb#(false(),x,y) -> c_7()
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
lt#(x,0()) -> c_10()
lt#(0(),s(x)) -> c_11()
lt#(s(x),s(y)) -> c_12(lt#(x,y))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
half#(0()) -> c_1()
half#(s(0())) -> c_2()
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(false(),x) -> c_5()
ifa#(true(),x) -> c_6(help#(x,1()))
ifb#(false(),x,y) -> c_7()
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
lt#(x,0()) -> c_10()
lt#(0(),s(x)) -> c_11()
lt#(s(x),s(y)) -> c_12(lt#(x,y))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
help(x,y) -> ifb(lt(y,x),x,y)
ifa(false(),x) -> logZeroError()
ifa(true(),x) -> help(x,1())
ifb(false(),x,y) -> y
ifb(true(),x,y) -> help(half(x),s(y))
logarithm(x) -> ifa(lt(0(),x),x)
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{half/1,help/2,ifa/2,ifb/3,logarithm/1,lt/2,half#/1,help#/2,ifa#/2,ifb#/3,logarithm#/1,lt#/2} / {0/0,1/0
,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half#,help#,ifa#,ifb#,logarithm#,lt#} and constructors {0
,1,false,logZeroError,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
half#(0()) -> c_1()
half#(s(0())) -> c_2()
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(false(),x) -> c_5()
ifa#(true(),x) -> c_6(help#(x,1()))
ifb#(false(),x,y) -> c_7()
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
lt#(x,0()) -> c_10()
lt#(0(),s(x)) -> c_11()
lt#(s(x),s(y)) -> c_12(lt#(x,y))
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
half#(0()) -> c_1()
half#(s(0())) -> c_2()
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(false(),x) -> c_5()
ifa#(true(),x) -> c_6(help#(x,1()))
ifb#(false(),x,y) -> c_7()
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
lt#(x,0()) -> c_10()
lt#(0(),s(x)) -> c_11()
lt#(s(x),s(y)) -> c_12(lt#(x,y))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{half/1,help/2,ifa/2,ifb/3,logarithm/1,lt/2,half#/1,help#/2,ifa#/2,ifb#/3,logarithm#/1,lt#/2} / {0/0,1/0
,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half#,help#,ifa#,ifb#,logarithm#,lt#} and constructors {0
,1,false,logZeroError,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,5,7,10,11}
by application of
Pre({1,2,5,7,10,11}) = {3,4,8,9,12}.
Here rules are labelled as follows:
1: half#(0()) -> c_1()
2: half#(s(0())) -> c_2()
3: half#(s(s(x))) -> c_3(half#(x))
4: help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
5: ifa#(false(),x) -> c_5()
6: ifa#(true(),x) -> c_6(help#(x,1()))
7: ifb#(false(),x,y) -> c_7()
8: ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
9: logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
10: lt#(x,0()) -> c_10()
11: lt#(0(),s(x)) -> c_11()
12: lt#(s(x),s(y)) -> c_12(lt#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(true(),x) -> c_6(help#(x,1()))
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
lt#(s(x),s(y)) -> c_12(lt#(x,y))
- Weak DPs:
half#(0()) -> c_1()
half#(s(0())) -> c_2()
ifa#(false(),x) -> c_5()
ifb#(false(),x,y) -> c_7()
lt#(x,0()) -> c_10()
lt#(0(),s(x)) -> c_11()
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{half/1,help/2,ifa/2,ifb/3,logarithm/1,lt/2,half#/1,help#/2,ifa#/2,ifb#/3,logarithm#/1,lt#/2} / {0/0,1/0
,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half#,help#,ifa#,ifb#,logarithm#,lt#} and constructors {0
,1,false,logZeroError,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:half#(s(s(x))) -> c_3(half#(x))
-->_1 half#(s(0())) -> c_2():8
-->_1 half#(0()) -> c_1():7
-->_1 half#(s(s(x))) -> c_3(half#(x)):1
2:S:help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
-->_2 lt#(s(x),s(y)) -> c_12(lt#(x,y)):6
-->_1 ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x)):4
-->_2 lt#(0(),s(x)) -> c_11():12
-->_2 lt#(x,0()) -> c_10():11
-->_1 ifb#(false(),x,y) -> c_7():10
3:S:ifa#(true(),x) -> c_6(help#(x,1()))
-->_1 help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x)):2
4:S:ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
-->_2 half#(s(0())) -> c_2():8
-->_2 half#(0()) -> c_1():7
-->_1 help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x)):2
-->_2 half#(s(s(x))) -> c_3(half#(x)):1
5:S:logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
-->_2 lt#(0(),s(x)) -> c_11():12
-->_2 lt#(x,0()) -> c_10():11
-->_1 ifa#(false(),x) -> c_5():9
-->_1 ifa#(true(),x) -> c_6(help#(x,1())):3
6:S:lt#(s(x),s(y)) -> c_12(lt#(x,y))
-->_1 lt#(0(),s(x)) -> c_11():12
-->_1 lt#(x,0()) -> c_10():11
-->_1 lt#(s(x),s(y)) -> c_12(lt#(x,y)):6
7:W:half#(0()) -> c_1()
8:W:half#(s(0())) -> c_2()
9:W:ifa#(false(),x) -> c_5()
10:W:ifb#(false(),x,y) -> c_7()
11:W:lt#(x,0()) -> c_10()
12:W:lt#(0(),s(x)) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: ifa#(false(),x) -> c_5()
10: ifb#(false(),x,y) -> c_7()
11: lt#(x,0()) -> c_10()
12: lt#(0(),s(x)) -> c_11()
7: half#(0()) -> c_1()
8: half#(s(0())) -> c_2()
* Step 5: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(true(),x) -> c_6(help#(x,1()))
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
lt#(s(x),s(y)) -> c_12(lt#(x,y))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{half/1,help/2,ifa/2,ifb/3,logarithm/1,lt/2,half#/1,help#/2,ifa#/2,ifb#/3,logarithm#/1,lt#/2} / {0/0,1/0
,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half#,help#,ifa#,ifb#,logarithm#,lt#} and constructors {0
,1,false,logZeroError,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:half#(s(s(x))) -> c_3(half#(x))
-->_1 half#(s(s(x))) -> c_3(half#(x)):1
2:S:help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
-->_2 lt#(s(x),s(y)) -> c_12(lt#(x,y)):6
-->_1 ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x)):4
3:S:ifa#(true(),x) -> c_6(help#(x,1()))
-->_1 help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x)):2
4:S:ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
-->_1 help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x)):2
-->_2 half#(s(s(x))) -> c_3(half#(x)):1
5:S:logarithm#(x) -> c_9(ifa#(lt(0(),x),x),lt#(0(),x))
-->_1 ifa#(true(),x) -> c_6(help#(x,1())):3
6:S:lt#(s(x),s(y)) -> c_12(lt#(x,y))
-->_1 lt#(s(x),s(y)) -> c_12(lt#(x,y)):6
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
logarithm#(x) -> c_9(ifa#(lt(0(),x),x))
* Step 6: NaturalMI MAYBE
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(true(),x) -> c_6(help#(x,1()))
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
logarithm#(x) -> c_9(ifa#(lt(0(),x),x))
lt#(s(x),s(y)) -> c_12(lt#(x,y))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{half/1,help/2,ifa/2,ifb/3,logarithm/1,lt/2,half#/1,help#/2,ifa#/2,ifb#/3,logarithm#/1,lt#/2} / {0/0,1/0
,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0,c_6/1,c_7/0,c_8/2,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half#,help#,ifa#,ifb#,logarithm#,lt#} and constructors {0
,1,false,logZeroError,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_3) = {1},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1,2},
uargs(c_9) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
{half#,help#,ifa#,ifb#,logarithm#,lt#}
TcT has computed the following interpretation:
p(0) = [1]
p(1) = [0]
p(false) = [8]
p(half) = [2]
p(help) = [1] x1 + [8] x2 + [4]
p(ifa) = [2] x1 + [1] x2 + [1]
p(ifb) = [1] x2 + [1] x3 + [1]
p(logZeroError) = [0]
p(logarithm) = [1] x1 + [1]
p(lt) = [0]
p(s) = [0]
p(true) = [0]
p(half#) = [0]
p(help#) = [8] x2 + [10]
p(ifa#) = [11]
p(ifb#) = [10]
p(logarithm#) = [14]
p(lt#) = [0]
p(c_1) = [8]
p(c_2) = [1]
p(c_3) = [4] x1 + [0]
p(c_4) = [1] x1 + [8] x2 + [0]
p(c_5) = [8]
p(c_6) = [1] x1 + [1]
p(c_7) = [1]
p(c_8) = [1] x1 + [8] x2 + [0]
p(c_9) = [1] x1 + [2]
p(c_10) = [1]
p(c_11) = [4]
p(c_12) = [8] x1 + [0]
Following rules are strictly oriented:
logarithm#(x) = [14]
> [13]
= c_9(ifa#(lt(0(),x),x))
Following rules are (at-least) weakly oriented:
half#(s(s(x))) = [0]
>= [0]
= c_3(half#(x))
help#(x,y) = [8] y + [10]
>= [10]
= c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(true(),x) = [11]
>= [11]
= c_6(help#(x,1()))
ifb#(true(),x,y) = [10]
>= [10]
= c_8(help#(half(x),s(y)),half#(x))
lt#(s(x),s(y)) = [0]
>= [0]
= c_12(lt#(x,y))
* Step 7: NaturalMI MAYBE
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifa#(true(),x) -> c_6(help#(x,1()))
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
lt#(s(x),s(y)) -> c_12(lt#(x,y))
- Weak DPs:
logarithm#(x) -> c_9(ifa#(lt(0(),x),x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{half/1,help/2,ifa/2,ifb/3,logarithm/1,lt/2,half#/1,help#/2,ifa#/2,ifb#/3,logarithm#/1,lt#/2} / {0/0,1/0
,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0,c_6/1,c_7/0,c_8/2,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half#,help#,ifa#,ifb#,logarithm#,lt#} and constructors {0
,1,false,logZeroError,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_3) = {1},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1,2},
uargs(c_9) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
{half#,help#,ifa#,ifb#,logarithm#,lt#}
TcT has computed the following interpretation:
p(0) = [0]
p(1) = [1]
p(false) = [0]
p(half) = [7]
p(help) = [2] x1 + [1] x2 + [0]
p(ifa) = [1] x1 + [1] x2 + [8]
p(ifb) = [8] x1 + [1] x3 + [1]
p(logZeroError) = [1]
p(logarithm) = [0]
p(lt) = [8] x2 + [0]
p(s) = [1]
p(true) = [0]
p(half#) = [0]
p(help#) = [0]
p(ifa#) = [15]
p(ifb#) = [0]
p(logarithm#) = [15]
p(lt#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [4] x1 + [0]
p(c_4) = [8] x1 + [4] x2 + [0]
p(c_5) = [4]
p(c_6) = [1] x1 + [11]
p(c_7) = [1]
p(c_8) = [8] x1 + [2] x2 + [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [2]
p(c_12) = [4] x1 + [0]
Following rules are strictly oriented:
ifa#(true(),x) = [15]
> [11]
= c_6(help#(x,1()))
Following rules are (at-least) weakly oriented:
half#(s(s(x))) = [0]
>= [0]
= c_3(half#(x))
help#(x,y) = [0]
>= [0]
= c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifb#(true(),x,y) = [0]
>= [0]
= c_8(help#(half(x),s(y)),half#(x))
logarithm#(x) = [15]
>= [15]
= c_9(ifa#(lt(0(),x),x))
lt#(s(x),s(y)) = [0]
>= [0]
= c_12(lt#(x,y))
* Step 8: Failure MAYBE
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_3(half#(x))
help#(x,y) -> c_4(ifb#(lt(y,x),x,y),lt#(y,x))
ifb#(true(),x,y) -> c_8(help#(half(x),s(y)),half#(x))
lt#(s(x),s(y)) -> c_12(lt#(x,y))
- Weak DPs:
ifa#(true(),x) -> c_6(help#(x,1()))
logarithm#(x) -> c_9(ifa#(lt(0(),x),x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{half/1,help/2,ifa/2,ifb/3,logarithm/1,lt/2,half#/1,help#/2,ifa#/2,ifb#/3,logarithm#/1,lt#/2} / {0/0,1/0
,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0,c_6/1,c_7/0,c_8/2,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half#,help#,ifa#,ifb#,logarithm#,lt#} and constructors {0
,1,false,logZeroError,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE