MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
if(false(),b,x,y) -> if2(b,x,y)
if(true(),b,x,y) -> log_undefined()
if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
if2(true(),x,s(y)) -> y
inc(0()) -> 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) -> log2(x,0())
log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
minus(x,0()) -> x
minus(0(),y) -> 0()
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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2} / {0/0,false/0,log_undefined/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if,if2,inc,le,log,log2,minus,quot} and constructors {0
,false,log_undefined,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
if#(false(),b,x,y) -> c_1(if2#(b,x,y))
if#(true(),b,x,y) -> c_2()
if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
if2#(true(),x,s(y)) -> c_4()
inc#(0()) -> c_5()
inc#(s(x)) -> c_6(inc#(x))
le#(0(),y) -> c_7()
le#(s(x),0()) -> c_8()
le#(s(x),s(y)) -> c_9(le#(x,y))
log#(x) -> c_10(log2#(x,0()))
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
minus#(x,0()) -> c_12()
minus#(0(),y) -> c_13()
minus#(s(x),s(y)) -> c_14(minus#(x,y))
quot#(0(),s(y)) -> c_15()
quot#(s(x),s(y)) -> c_16(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(if2#(b,x,y))
if#(true(),b,x,y) -> c_2()
if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
if2#(true(),x,s(y)) -> c_4()
inc#(0()) -> c_5()
inc#(s(x)) -> c_6(inc#(x))
le#(0(),y) -> c_7()
le#(s(x),0()) -> c_8()
le#(s(x),s(y)) -> c_9(le#(x,y))
log#(x) -> c_10(log2#(x,0()))
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
minus#(x,0()) -> c_12()
minus#(0(),y) -> c_13()
minus#(s(x),s(y)) -> c_14(minus#(x,y))
quot#(0(),s(y)) -> c_15()
quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if(false(),b,x,y) -> if2(b,x,y)
if(true(),b,x,y) -> log_undefined()
if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
if2(true(),x,s(y)) -> y
inc(0()) -> 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) -> log2(x,0())
log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
minus(x,0()) -> x
minus(0(),y) -> 0()
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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2
,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1
,c_10/1,c_11/4,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus#
,quot#} and constructors {0,false,log_undefined,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
inc(0()) -> 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(0(),y) -> 0()
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(if2#(b,x,y))
if#(true(),b,x,y) -> c_2()
if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
if2#(true(),x,s(y)) -> c_4()
inc#(0()) -> c_5()
inc#(s(x)) -> c_6(inc#(x))
le#(0(),y) -> c_7()
le#(s(x),0()) -> c_8()
le#(s(x),s(y)) -> c_9(le#(x,y))
log#(x) -> c_10(log2#(x,0()))
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
minus#(x,0()) -> c_12()
minus#(0(),y) -> c_13()
minus#(s(x),s(y)) -> c_14(minus#(x,y))
quot#(0(),s(y)) -> c_15()
quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
if#(false(),b,x,y) -> c_1(if2#(b,x,y))
if#(true(),b,x,y) -> c_2()
if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
if2#(true(),x,s(y)) -> c_4()
inc#(0()) -> c_5()
inc#(s(x)) -> c_6(inc#(x))
le#(0(),y) -> c_7()
le#(s(x),0()) -> c_8()
le#(s(x),s(y)) -> c_9(le#(x,y))
log#(x) -> c_10(log2#(x,0()))
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
minus#(x,0()) -> c_12()
minus#(0(),y) -> c_13()
minus#(s(x),s(y)) -> c_14(minus#(x,y))
quot#(0(),s(y)) -> c_15()
quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
inc(0()) -> 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(0(),y) -> 0()
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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2
,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1
,c_10/1,c_11/4,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus#
,quot#} and constructors {0,false,log_undefined,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4,5,7,8,12,13,15}
by application of
Pre({2,4,5,7,8,12,13,15}) = {1,3,6,9,11,14,16}.
Here rules are labelled as follows:
1: if#(false(),b,x,y) -> c_1(if2#(b,x,y))
2: if#(true(),b,x,y) -> c_2()
3: if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
4: if2#(true(),x,s(y)) -> c_4()
5: inc#(0()) -> c_5()
6: inc#(s(x)) -> c_6(inc#(x))
7: le#(0(),y) -> c_7()
8: le#(s(x),0()) -> c_8()
9: le#(s(x),s(y)) -> c_9(le#(x,y))
10: log#(x) -> c_10(log2#(x,0()))
11: log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
12: minus#(x,0()) -> c_12()
13: minus#(0(),y) -> c_13()
14: minus#(s(x),s(y)) -> c_14(minus#(x,y))
15: quot#(0(),s(y)) -> c_15()
16: quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
if#(false(),b,x,y) -> c_1(if2#(b,x,y))
if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
inc#(s(x)) -> c_6(inc#(x))
le#(s(x),s(y)) -> c_9(le#(x,y))
log#(x) -> c_10(log2#(x,0()))
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
minus#(s(x),s(y)) -> c_14(minus#(x,y))
quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
if#(true(),b,x,y) -> c_2()
if2#(true(),x,s(y)) -> c_4()
inc#(0()) -> c_5()
le#(0(),y) -> c_7()
le#(s(x),0()) -> c_8()
minus#(x,0()) -> c_12()
minus#(0(),y) -> c_13()
quot#(0(),s(y)) -> c_15()
- Weak TRS:
inc(0()) -> 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(0(),y) -> 0()
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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2
,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1
,c_10/1,c_11/4,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus#
,quot#} and constructors {0,false,log_undefined,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if#(false(),b,x,y) -> c_1(if2#(b,x,y))
-->_1 if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))):2
-->_1 if2#(true(),x,s(y)) -> c_4():10
2:S:if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8
-->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)):6
-->_2 quot#(0(),s(y)) -> c_15():16
3:S:inc#(s(x)) -> c_6(inc#(x))
-->_1 inc#(0()) -> c_5():11
-->_1 inc#(s(x)) -> c_6(inc#(x)):3
4:S:le#(s(x),s(y)) -> c_9(le#(x,y))
-->_1 le#(s(x),0()) -> c_8():13
-->_1 le#(0(),y) -> c_7():12
-->_1 le#(s(x),s(y)) -> c_9(le#(x,y)):4
5:S:log#(x) -> c_10(log2#(x,0()))
-->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)):6
6:S:log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
-->_2 le#(s(x),0()) -> c_8():13
-->_3 le#(0(),y) -> c_7():12
-->_2 le#(0(),y) -> c_7():12
-->_4 inc#(0()) -> c_5():11
-->_1 if#(true(),b,x,y) -> c_2():9
-->_3 le#(s(x),s(y)) -> c_9(le#(x,y)):4
-->_4 inc#(s(x)) -> c_6(inc#(x)):3
-->_1 if#(false(),b,x,y) -> c_1(if2#(b,x,y)):1
7:S:minus#(s(x),s(y)) -> c_14(minus#(x,y))
-->_1 minus#(0(),y) -> c_13():15
-->_1 minus#(x,0()) -> c_12():14
-->_1 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7
8:S:quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(0(),s(y)) -> c_15():16
-->_2 minus#(0(),y) -> c_13():15
-->_2 minus#(x,0()) -> c_12():14
-->_1 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8
-->_2 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7
9:W:if#(true(),b,x,y) -> c_2()
10:W:if2#(true(),x,s(y)) -> c_4()
11:W:inc#(0()) -> c_5()
12:W:le#(0(),y) -> c_7()
13:W:le#(s(x),0()) -> c_8()
14:W:minus#(x,0()) -> c_12()
15:W:minus#(0(),y) -> c_13()
16:W:quot#(0(),s(y)) -> c_15()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: if2#(true(),x,s(y)) -> c_4()
9: if#(true(),b,x,y) -> c_2()
11: inc#(0()) -> c_5()
12: le#(0(),y) -> c_7()
13: le#(s(x),0()) -> c_8()
14: minus#(x,0()) -> c_12()
15: minus#(0(),y) -> c_13()
16: quot#(0(),s(y)) -> c_15()
* Step 5: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
if#(false(),b,x,y) -> c_1(if2#(b,x,y))
if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
inc#(s(x)) -> c_6(inc#(x))
le#(s(x),s(y)) -> c_9(le#(x,y))
log#(x) -> c_10(log2#(x,0()))
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
minus#(s(x),s(y)) -> c_14(minus#(x,y))
quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
inc(0()) -> 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(0(),y) -> 0()
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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2
,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1
,c_10/1,c_11/4,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus#
,quot#} and constructors {0,false,log_undefined,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:if#(false(),b,x,y) -> c_1(if2#(b,x,y))
-->_1 if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))):2
2:S:if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8
-->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)):6
3:S:inc#(s(x)) -> c_6(inc#(x))
-->_1 inc#(s(x)) -> c_6(inc#(x)):3
4:S:le#(s(x),s(y)) -> c_9(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_9(le#(x,y)):4
5:S:log#(x) -> c_10(log2#(x,0()))
-->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)):6
6:S:log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y))
-->_3 le#(s(x),s(y)) -> c_9(le#(x,y)):4
-->_4 inc#(s(x)) -> c_6(inc#(x)):3
-->_1 if#(false(),b,x,y) -> c_1(if2#(b,x,y)):1
7:S:minus#(s(x),s(y)) -> c_14(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7
8:S:quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8
-->_2 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y))
* Step 6: RemoveHeads MAYBE
+ Considered Problem:
- Strict DPs:
if#(false(),b,x,y) -> c_1(if2#(b,x,y))
if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
inc#(s(x)) -> c_6(inc#(x))
le#(s(x),s(y)) -> c_9(le#(x,y))
log#(x) -> c_10(log2#(x,0()))
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y))
minus#(s(x),s(y)) -> c_14(minus#(x,y))
quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
inc(0()) -> 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(0(),y) -> 0()
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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2
,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1
,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus#
,quot#} and constructors {0,false,log_undefined,s,true}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:if#(false(),b,x,y) -> c_1(if2#(b,x,y))
-->_1 if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))):2
2:S:if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8
-->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y)):6
3:S:inc#(s(x)) -> c_6(inc#(x))
-->_1 inc#(s(x)) -> c_6(inc#(x)):3
4:S:le#(s(x),s(y)) -> c_9(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_9(le#(x,y)):4
5:S:log#(x) -> c_10(log2#(x,0()))
-->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y)):6
6:S:log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y))
-->_2 le#(s(x),s(y)) -> c_9(le#(x,y)):4
-->_3 inc#(s(x)) -> c_6(inc#(x)):3
-->_1 if#(false(),b,x,y) -> c_1(if2#(b,x,y)):1
7:S:minus#(s(x),s(y)) -> c_14(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7
8:S:quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8
-->_2 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7
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,log#(x) -> c_10(log2#(x,0())))]
* Step 7: Failure MAYBE
+ Considered Problem:
- Strict DPs:
if#(false(),b,x,y) -> c_1(if2#(b,x,y))
if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0()))))
inc#(s(x)) -> c_6(inc#(x))
le#(s(x),s(y)) -> c_9(le#(x,y))
log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y))
minus#(s(x),s(y)) -> c_14(minus#(x,y))
quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
inc(0()) -> 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(0(),y) -> 0()
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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2
,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1
,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus#
,quot#} and constructors {0,false,log_undefined,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE