MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
if(false(),b,x,y) -> logZeroError()
if(true(),false(),x,s(y)) -> y
if(true(),true(),x,y) -> logIter(x,y)
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
log(x) -> logIter(x,0())
logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2} / {0/0,false/0,logZeroError/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if,inc,le,log,logIter,minus,quot} and constructors {0
,false,logZeroError,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
if#(false(),b,x,y) -> c_1()
if#(true(),false(),x,s(y)) -> c_2()
if#(true(),true(),x,y) -> c_3(logIter#(x,y))
inc#(0()) -> c_4()
inc#(s(x)) -> c_5(inc#(x))
le#(0(),y) -> c_6()
le#(s(x),0()) -> c_7()
le#(s(x),s(y)) -> c_8(le#(x,y))
log#(x) -> c_9(logIter#(x,0()))
logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
minus#(x,0()) -> c_11()
minus#(s(x),s(y)) -> c_12(minus#(x,y))
quot#(0(),s(y)) -> c_13()
quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
if#(false(),b,x,y) -> c_1()
if#(true(),false(),x,s(y)) -> c_2()
if#(true(),true(),x,y) -> c_3(logIter#(x,y))
inc#(0()) -> c_4()
inc#(s(x)) -> c_5(inc#(x))
le#(0(),y) -> c_6()
le#(s(x),0()) -> c_7()
le#(s(x),s(y)) -> c_8(le#(x,y))
log#(x) -> c_9(logIter#(x,0()))
logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
minus#(x,0()) -> c_11()
minus#(s(x),s(y)) -> c_12(minus#(x,y))
quot#(0(),s(y)) -> c_13()
quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if(false(),b,x,y) -> logZeroError()
if(true(),false(),x,s(y)) -> y
if(true(),true(),x,y) -> logIter(x,y)
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
log(x) -> logIter(x,0())
logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2
,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1
,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus#
,quot#} and constructors {0,false,logZeroError,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
if#(false(),b,x,y) -> c_1()
if#(true(),false(),x,s(y)) -> c_2()
if#(true(),true(),x,y) -> c_3(logIter#(x,y))
inc#(0()) -> c_4()
inc#(s(x)) -> c_5(inc#(x))
le#(0(),y) -> c_6()
le#(s(x),0()) -> c_7()
le#(s(x),s(y)) -> c_8(le#(x,y))
log#(x) -> c_9(logIter#(x,0()))
logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
minus#(x,0()) -> c_11()
minus#(s(x),s(y)) -> c_12(minus#(x,y))
quot#(0(),s(y)) -> c_13()
quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
if#(false(),b,x,y) -> c_1()
if#(true(),false(),x,s(y)) -> c_2()
if#(true(),true(),x,y) -> c_3(logIter#(x,y))
inc#(0()) -> c_4()
inc#(s(x)) -> c_5(inc#(x))
le#(0(),y) -> c_6()
le#(s(x),0()) -> c_7()
le#(s(x),s(y)) -> c_8(le#(x,y))
log#(x) -> c_9(logIter#(x,0()))
logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
minus#(x,0()) -> c_11()
minus#(s(x),s(y)) -> c_12(minus#(x,y))
quot#(0(),s(y)) -> c_13()
quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2
,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1
,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus#
,quot#} and constructors {0,false,logZeroError,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,4,6,7,11,13}
by application of
Pre({1,2,4,6,7,11,13}) = {5,8,10,12,14}.
Here rules are labelled as follows:
1: if#(false(),b,x,y) -> c_1()
2: if#(true(),false(),x,s(y)) -> c_2()
3: if#(true(),true(),x,y) -> c_3(logIter#(x,y))
4: inc#(0()) -> c_4()
5: inc#(s(x)) -> c_5(inc#(x))
6: le#(0(),y) -> c_6()
7: le#(s(x),0()) -> c_7()
8: le#(s(x),s(y)) -> c_8(le#(x,y))
9: log#(x) -> c_9(logIter#(x,0()))
10: logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
11: minus#(x,0()) -> c_11()
12: minus#(s(x),s(y)) -> c_12(minus#(x,y))
13: quot#(0(),s(y)) -> c_13()
14: quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
if#(true(),true(),x,y) -> c_3(logIter#(x,y))
inc#(s(x)) -> c_5(inc#(x))
le#(s(x),s(y)) -> c_8(le#(x,y))
log#(x) -> c_9(logIter#(x,0()))
logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
minus#(s(x),s(y)) -> c_12(minus#(x,y))
quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
if#(false(),b,x,y) -> c_1()
if#(true(),false(),x,s(y)) -> c_2()
inc#(0()) -> c_4()
le#(0(),y) -> c_6()
le#(s(x),0()) -> c_7()
minus#(x,0()) -> c_11()
quot#(0(),s(y)) -> c_13()
- Weak TRS:
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2
,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1
,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus#
,quot#} and constructors {0,false,logZeroError,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if#(true(),true(),x,y) -> c_3(logIter#(x,y))
-->_1 logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y)):5
2:S:inc#(s(x)) -> c_5(inc#(x))
-->_1 inc#(0()) -> c_4():10
-->_1 inc#(s(x)) -> c_5(inc#(x)):2
3:S:le#(s(x),s(y)) -> c_8(le#(x,y))
-->_1 le#(s(x),0()) -> c_7():12
-->_1 le#(0(),y) -> c_6():11
-->_1 le#(s(x),s(y)) -> c_8(le#(x,y)):3
4:S:log#(x) -> c_9(logIter#(x,0()))
-->_1 logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y)):5
5:S:logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
-->_4 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):7
-->_4 quot#(0(),s(y)) -> c_13():14
-->_3 le#(s(x),0()) -> c_7():12
-->_2 le#(s(x),0()) -> c_7():12
-->_5 inc#(0()) -> c_4():10
-->_1 if#(true(),false(),x,s(y)) -> c_2():9
-->_1 if#(false(),b,x,y) -> c_1():8
-->_3 le#(s(x),s(y)) -> c_8(le#(x,y)):3
-->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):3
-->_5 inc#(s(x)) -> c_5(inc#(x)):2
-->_1 if#(true(),true(),x,y) -> c_3(logIter#(x,y)):1
6:S:minus#(s(x),s(y)) -> c_12(minus#(x,y))
-->_1 minus#(x,0()) -> c_11():13
-->_1 minus#(s(x),s(y)) -> c_12(minus#(x,y)):6
7:S:quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(0(),s(y)) -> c_13():14
-->_2 minus#(x,0()) -> c_11():13
-->_1 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):7
-->_2 minus#(s(x),s(y)) -> c_12(minus#(x,y)):6
8:W:if#(false(),b,x,y) -> c_1()
9:W:if#(true(),false(),x,s(y)) -> c_2()
10:W:inc#(0()) -> c_4()
11:W:le#(0(),y) -> c_6()
12:W:le#(s(x),0()) -> c_7()
13:W:minus#(x,0()) -> c_11()
14:W:quot#(0(),s(y)) -> c_13()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
11: le#(0(),y) -> c_6()
8: if#(false(),b,x,y) -> c_1()
9: if#(true(),false(),x,s(y)) -> c_2()
10: inc#(0()) -> c_4()
12: le#(s(x),0()) -> c_7()
13: minus#(x,0()) -> c_11()
14: quot#(0(),s(y)) -> c_13()
* Step 5: RemoveHeads MAYBE
+ Considered Problem:
- Strict DPs:
if#(true(),true(),x,y) -> c_3(logIter#(x,y))
inc#(s(x)) -> c_5(inc#(x))
le#(s(x),s(y)) -> c_8(le#(x,y))
log#(x) -> c_9(logIter#(x,0()))
logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
minus#(s(x),s(y)) -> c_12(minus#(x,y))
quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2
,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1
,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus#
,quot#} and constructors {0,false,logZeroError,s,true}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:if#(true(),true(),x,y) -> c_3(logIter#(x,y))
-->_1 logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y)):5
2:S:inc#(s(x)) -> c_5(inc#(x))
-->_1 inc#(s(x)) -> c_5(inc#(x)):2
3:S:le#(s(x),s(y)) -> c_8(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_8(le#(x,y)):3
4:S:log#(x) -> c_9(logIter#(x,0()))
-->_1 logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y)):5
5:S:logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
-->_4 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):7
-->_3 le#(s(x),s(y)) -> c_8(le#(x,y)):3
-->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):3
-->_5 inc#(s(x)) -> c_5(inc#(x)):2
-->_1 if#(true(),true(),x,y) -> c_3(logIter#(x,y)):1
6:S:minus#(s(x),s(y)) -> c_12(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_12(minus#(x,y)):6
7:S:quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):7
-->_2 minus#(s(x),s(y)) -> c_12(minus#(x,y)):6
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).
[(4,log#(x) -> c_9(logIter#(x,0())))]
* Step 6: Failure MAYBE
+ Considered Problem:
- Strict DPs:
if#(true(),true(),x,y) -> c_3(logIter#(x,y))
inc#(s(x)) -> c_5(inc#(x))
le#(s(x),s(y)) -> c_8(le#(x,y))
logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
,le#(s(0()),x)
,le#(s(s(0())),x)
,quot#(x,s(s(0())))
,inc#(y))
minus#(s(x),s(y)) -> c_12(minus#(x,y))
quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2
,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1
,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus#
,quot#} and constructors {0,false,logZeroError,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE