MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
id(0()) -> 0()
id(s(x)) -> s(id(x))
if2(false(),b2,x,y) -> if3(b2,x,y)
if2(true(),b2,x,y) -> 0()
if3(false(),x,y) -> x
if3(true(),x,y) -> mod(minus(x,y),s(y))
if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
if_mod(true(),b1,b2,x,y) -> 0()
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)
mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
zero(0()) -> true()
zero(s(x)) -> false()
- Signature:
{id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {id,if2,if3,if_mod,le,minus,mod,zero} and constructors {0
,false,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
id#(0()) -> c_1()
id#(s(x)) -> c_2(id#(x))
if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
if2#(true(),b2,x,y) -> c_4()
if3#(false(),x,y) -> c_5()
if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
if_mod#(true(),b1,b2,x,y) -> c_8()
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
le#(s(x),s(y)) -> c_11(le#(x,y))
minus#(x,0()) -> c_12()
minus#(s(x),s(y)) -> c_13(minus#(x,y))
mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
zero#(0()) -> c_15()
zero#(s(x)) -> c_16()
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
id#(0()) -> c_1()
id#(s(x)) -> c_2(id#(x))
if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
if2#(true(),b2,x,y) -> c_4()
if3#(false(),x,y) -> c_5()
if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
if_mod#(true(),b1,b2,x,y) -> c_8()
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
le#(s(x),s(y)) -> c_11(le#(x,y))
minus#(x,0()) -> c_12()
minus#(s(x),s(y)) -> c_13(minus#(x,y))
mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
zero#(0()) -> c_15()
zero#(s(x)) -> c_16()
- Weak TRS:
id(0()) -> 0()
id(s(x)) -> s(id(x))
if2(false(),b2,x,y) -> if3(b2,x,y)
if2(true(),b2,x,y) -> 0()
if3(false(),x,y) -> x
if3(true(),x,y) -> mod(minus(x,y),s(y))
if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
if_mod(true(),b1,b2,x,y) -> 0()
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)
mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
zero(0()) -> true()
zero(s(x)) -> false()
- Signature:
{id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
,c_12/0,c_13/1,c_14/6,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
,zero#} and constructors {0,false,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
id(0()) -> 0()
id(s(x)) -> s(id(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)
zero(0()) -> true()
zero(s(x)) -> false()
id#(0()) -> c_1()
id#(s(x)) -> c_2(id#(x))
if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
if2#(true(),b2,x,y) -> c_4()
if3#(false(),x,y) -> c_5()
if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
if_mod#(true(),b1,b2,x,y) -> c_8()
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
le#(s(x),s(y)) -> c_11(le#(x,y))
minus#(x,0()) -> c_12()
minus#(s(x),s(y)) -> c_13(minus#(x,y))
mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
zero#(0()) -> c_15()
zero#(s(x)) -> c_16()
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
id#(0()) -> c_1()
id#(s(x)) -> c_2(id#(x))
if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
if2#(true(),b2,x,y) -> c_4()
if3#(false(),x,y) -> c_5()
if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
if_mod#(true(),b1,b2,x,y) -> c_8()
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
le#(s(x),s(y)) -> c_11(le#(x,y))
minus#(x,0()) -> c_12()
minus#(s(x),s(y)) -> c_13(minus#(x,y))
mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
zero#(0()) -> c_15()
zero#(s(x)) -> c_16()
- Weak TRS:
id(0()) -> 0()
id(s(x)) -> s(id(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)
zero(0()) -> true()
zero(s(x)) -> false()
- Signature:
{id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
,c_12/0,c_13/1,c_14/6,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
,zero#} and constructors {0,false,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,4,5,8,9,10,12,15,16}
by application of
Pre({1,4,5,8,9,10,12,15,16}) = {2,3,6,7,11,13,14}.
Here rules are labelled as follows:
1: id#(0()) -> c_1()
2: id#(s(x)) -> c_2(id#(x))
3: if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
4: if2#(true(),b2,x,y) -> c_4()
5: if3#(false(),x,y) -> c_5()
6: if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
7: if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
8: if_mod#(true(),b1,b2,x,y) -> c_8()
9: le#(0(),y) -> c_9()
10: le#(s(x),0()) -> c_10()
11: le#(s(x),s(y)) -> c_11(le#(x,y))
12: minus#(x,0()) -> c_12()
13: minus#(s(x),s(y)) -> c_13(minus#(x,y))
14: mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
15: zero#(0()) -> c_15()
16: zero#(s(x)) -> c_16()
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
id#(s(x)) -> c_2(id#(x))
if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
le#(s(x),s(y)) -> c_11(le#(x,y))
minus#(s(x),s(y)) -> c_13(minus#(x,y))
mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
- Weak DPs:
id#(0()) -> c_1()
if2#(true(),b2,x,y) -> c_4()
if3#(false(),x,y) -> c_5()
if_mod#(true(),b1,b2,x,y) -> c_8()
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
minus#(x,0()) -> c_12()
zero#(0()) -> c_15()
zero#(s(x)) -> c_16()
- Weak TRS:
id(0()) -> 0()
id(s(x)) -> s(id(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)
zero(0()) -> true()
zero(s(x)) -> false()
- Signature:
{id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
,c_12/0,c_13/1,c_14/6,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
,zero#} and constructors {0,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:id#(s(x)) -> c_2(id#(x))
-->_1 id#(0()) -> c_1():8
-->_1 id#(s(x)) -> c_2(id#(x)):1
2:S:if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
-->_1 if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 if3#(false(),x,y) -> c_5():10
3:S:if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
,zero#(x)
,zero#(y)
,le#(y,x)
,id#(x)
,id#(y)):7
-->_2 minus#(s(x),s(y)) -> c_13(minus#(x,y)):6
-->_2 minus#(x,0()) -> c_12():14
4:S:if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
-->_1 if2#(true(),b2,x,y) -> c_4():9
-->_1 if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y)):2
5:S:le#(s(x),s(y)) -> c_11(le#(x,y))
-->_1 le#(s(x),0()) -> c_10():13
-->_1 le#(0(),y) -> c_9():12
-->_1 le#(s(x),s(y)) -> c_11(le#(x,y)):5
6:S:minus#(s(x),s(y)) -> c_13(minus#(x,y))
-->_1 minus#(x,0()) -> c_12():14
-->_1 minus#(s(x),s(y)) -> c_13(minus#(x,y)):6
7:S:mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
-->_3 zero#(s(x)) -> c_16():16
-->_2 zero#(s(x)) -> c_16():16
-->_3 zero#(0()) -> c_15():15
-->_2 zero#(0()) -> c_15():15
-->_4 le#(s(x),0()) -> c_10():13
-->_4 le#(0(),y) -> c_9():12
-->_1 if_mod#(true(),b1,b2,x,y) -> c_8():11
-->_6 id#(0()) -> c_1():8
-->_5 id#(0()) -> c_1():8
-->_4 le#(s(x),s(y)) -> c_11(le#(x,y)):5
-->_1 if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y)):4
-->_6 id#(s(x)) -> c_2(id#(x)):1
-->_5 id#(s(x)) -> c_2(id#(x)):1
8:W:id#(0()) -> c_1()
9:W:if2#(true(),b2,x,y) -> c_4()
10:W:if3#(false(),x,y) -> c_5()
11:W:if_mod#(true(),b1,b2,x,y) -> c_8()
12:W:le#(0(),y) -> c_9()
13:W:le#(s(x),0()) -> c_10()
14:W:minus#(x,0()) -> c_12()
15:W:zero#(0()) -> c_15()
16:W:zero#(s(x)) -> c_16()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: if3#(false(),x,y) -> c_5()
14: minus#(x,0()) -> c_12()
9: if2#(true(),b2,x,y) -> c_4()
11: if_mod#(true(),b1,b2,x,y) -> c_8()
12: le#(0(),y) -> c_9()
13: le#(s(x),0()) -> c_10()
15: zero#(0()) -> c_15()
16: zero#(s(x)) -> c_16()
8: id#(0()) -> c_1()
* Step 5: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
id#(s(x)) -> c_2(id#(x))
if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
le#(s(x),s(y)) -> c_11(le#(x,y))
minus#(s(x),s(y)) -> c_13(minus#(x,y))
mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
- Weak TRS:
id(0()) -> 0()
id(s(x)) -> s(id(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)
zero(0()) -> true()
zero(s(x)) -> false()
- Signature:
{id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
,c_12/0,c_13/1,c_14/6,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
,zero#} and constructors {0,false,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:id#(s(x)) -> c_2(id#(x))
-->_1 id#(s(x)) -> c_2(id#(x)):1
2:S:if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
-->_1 if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y)):3
3:S:if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
,zero#(x)
,zero#(y)
,le#(y,x)
,id#(x)
,id#(y)):7
-->_2 minus#(s(x),s(y)) -> c_13(minus#(x,y)):6
4:S:if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
-->_1 if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y)):2
5:S:le#(s(x),s(y)) -> c_11(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_11(le#(x,y)):5
6:S:minus#(s(x),s(y)) -> c_13(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_13(minus#(x,y)):6
7:S:mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
-->_4 le#(s(x),s(y)) -> c_11(le#(x,y)):5
-->_1 if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y)):4
-->_6 id#(s(x)) -> c_2(id#(x)):1
-->_5 id#(s(x)) -> c_2(id#(x)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),le#(y,x),id#(x),id#(y))
* Step 6: Failure MAYBE
+ Considered Problem:
- Strict DPs:
id#(s(x)) -> c_2(id#(x))
if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
le#(s(x),s(y)) -> c_11(le#(x,y))
minus#(s(x),s(y)) -> c_13(minus#(x,y))
mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),le#(y,x),id#(x),id#(y))
- Weak TRS:
id(0()) -> 0()
id(s(x)) -> s(id(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)
zero(0()) -> true()
zero(s(x)) -> false()
- Signature:
{id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
,c_12/0,c_13/1,c_14/4,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
,zero#} and constructors {0,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE