MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,xs) -> min(cons(y,xs))
if1(true(),x,y,xs) -> min(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,rm(x,xs))
if2(true(),x,y,xs) -> rm(x,xs)
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
min(cons(x,cons(y,xs))) -> if1(le(x,y),x,y,xs)
min(cons(x,nil())) -> x
min(nil()) -> 0()
minsort(cons(x,xs)) -> cons(min(cons(x,xs)),minsort(rm(min(cons(x,xs)),cons(x,xs))))
minsort(nil()) -> nil()
rm(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
rm(x,nil()) -> nil()
- Signature:
{eq/2,if1/4,if2/4,le/2,min/1,minsort/1,rm/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,if1,if2,le,min,minsort,rm} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
eq#(0(),0()) -> c_1()
eq#(0(),s(y)) -> c_2()
eq#(s(x),0()) -> c_3()
eq#(s(x),s(y)) -> c_4(eq#(x,y))
if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs)))
if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs)))
if2#(false(),x,y,xs) -> c_7(rm#(x,xs))
if2#(true(),x,y,xs) -> c_8(rm#(x,xs))
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
le#(s(x),s(y)) -> c_11(le#(x,y))
min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y))
min#(cons(x,nil())) -> c_13()
min#(nil()) -> c_14()
minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs)))
minsort#(nil()) -> c_16()
rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y))
rm#(x,nil()) -> c_18()
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
eq#(0(),0()) -> c_1()
eq#(0(),s(y)) -> c_2()
eq#(s(x),0()) -> c_3()
eq#(s(x),s(y)) -> c_4(eq#(x,y))
if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs)))
if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs)))
if2#(false(),x,y,xs) -> c_7(rm#(x,xs))
if2#(true(),x,y,xs) -> c_8(rm#(x,xs))
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
le#(s(x),s(y)) -> c_11(le#(x,y))
min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y))
min#(cons(x,nil())) -> c_13()
min#(nil()) -> c_14()
minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs)))
minsort#(nil()) -> c_16()
rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y))
rm#(x,nil()) -> c_18()
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,xs) -> min(cons(y,xs))
if1(true(),x,y,xs) -> min(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,rm(x,xs))
if2(true(),x,y,xs) -> rm(x,xs)
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
min(cons(x,cons(y,xs))) -> if1(le(x,y),x,y,xs)
min(cons(x,nil())) -> x
min(nil()) -> 0()
minsort(cons(x,xs)) -> cons(min(cons(x,xs)),minsort(rm(min(cons(x,xs)),cons(x,xs))))
minsort(nil()) -> nil()
rm(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
rm(x,nil()) -> nil()
- Signature:
{eq/2,if1/4,if2/4,le/2,min/1,minsort/1,rm/2,eq#/2,if1#/4,if2#/4,le#/2,min#/1,minsort#/1,rm#/2} / {0/0,cons/2
,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0
,c_14/0,c_15/4,c_16/0,c_17/2,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq#,if1#,if2#,le#,min#,minsort#,rm#} and constructors {0
,cons,false,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,xs) -> min(cons(y,xs))
if1(true(),x,y,xs) -> min(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,rm(x,xs))
if2(true(),x,y,xs) -> rm(x,xs)
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
min(cons(x,cons(y,xs))) -> if1(le(x,y),x,y,xs)
min(cons(x,nil())) -> x
rm(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
rm(x,nil()) -> nil()
eq#(0(),0()) -> c_1()
eq#(0(),s(y)) -> c_2()
eq#(s(x),0()) -> c_3()
eq#(s(x),s(y)) -> c_4(eq#(x,y))
if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs)))
if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs)))
if2#(false(),x,y,xs) -> c_7(rm#(x,xs))
if2#(true(),x,y,xs) -> c_8(rm#(x,xs))
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
le#(s(x),s(y)) -> c_11(le#(x,y))
min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y))
min#(cons(x,nil())) -> c_13()
min#(nil()) -> c_14()
minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs)))
minsort#(nil()) -> c_16()
rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y))
rm#(x,nil()) -> c_18()
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
eq#(0(),0()) -> c_1()
eq#(0(),s(y)) -> c_2()
eq#(s(x),0()) -> c_3()
eq#(s(x),s(y)) -> c_4(eq#(x,y))
if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs)))
if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs)))
if2#(false(),x,y,xs) -> c_7(rm#(x,xs))
if2#(true(),x,y,xs) -> c_8(rm#(x,xs))
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
le#(s(x),s(y)) -> c_11(le#(x,y))
min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y))
min#(cons(x,nil())) -> c_13()
min#(nil()) -> c_14()
minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs)))
minsort#(nil()) -> c_16()
rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y))
rm#(x,nil()) -> c_18()
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,xs) -> min(cons(y,xs))
if1(true(),x,y,xs) -> min(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,rm(x,xs))
if2(true(),x,y,xs) -> rm(x,xs)
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
min(cons(x,cons(y,xs))) -> if1(le(x,y),x,y,xs)
min(cons(x,nil())) -> x
rm(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
rm(x,nil()) -> nil()
- Signature:
{eq/2,if1/4,if2/4,le/2,min/1,minsort/1,rm/2,eq#/2,if1#/4,if2#/4,le#/2,min#/1,minsort#/1,rm#/2} / {0/0,cons/2
,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0
,c_14/0,c_15/4,c_16/0,c_17/2,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq#,if1#,if2#,le#,min#,minsort#,rm#} and constructors {0
,cons,false,nil,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,3,9,10,13,14,16,18}
by application of
Pre({1,2,3,9,10,13,14,16,18}) = {4,5,6,7,8,11,12,15,17}.
Here rules are labelled as follows:
1: eq#(0(),0()) -> c_1()
2: eq#(0(),s(y)) -> c_2()
3: eq#(s(x),0()) -> c_3()
4: eq#(s(x),s(y)) -> c_4(eq#(x,y))
5: if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs)))
6: if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs)))
7: if2#(false(),x,y,xs) -> c_7(rm#(x,xs))
8: if2#(true(),x,y,xs) -> c_8(rm#(x,xs))
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: min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y))
13: min#(cons(x,nil())) -> c_13()
14: min#(nil()) -> c_14()
15: minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs)))
16: minsort#(nil()) -> c_16()
17: rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y))
18: rm#(x,nil()) -> c_18()
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
eq#(s(x),s(y)) -> c_4(eq#(x,y))
if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs)))
if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs)))
if2#(false(),x,y,xs) -> c_7(rm#(x,xs))
if2#(true(),x,y,xs) -> c_8(rm#(x,xs))
le#(s(x),s(y)) -> c_11(le#(x,y))
min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y))
minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs)))
rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y))
- Weak DPs:
eq#(0(),0()) -> c_1()
eq#(0(),s(y)) -> c_2()
eq#(s(x),0()) -> c_3()
le#(0(),y) -> c_9()
le#(s(x),0()) -> c_10()
min#(cons(x,nil())) -> c_13()
min#(nil()) -> c_14()
minsort#(nil()) -> c_16()
rm#(x,nil()) -> c_18()
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,xs) -> min(cons(y,xs))
if1(true(),x,y,xs) -> min(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,rm(x,xs))
if2(true(),x,y,xs) -> rm(x,xs)
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
min(cons(x,cons(y,xs))) -> if1(le(x,y),x,y,xs)
min(cons(x,nil())) -> x
rm(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
rm(x,nil()) -> nil()
- Signature:
{eq/2,if1/4,if2/4,le/2,min/1,minsort/1,rm/2,eq#/2,if1#/4,if2#/4,le#/2,min#/1,minsort#/1,rm#/2} / {0/0,cons/2
,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0
,c_14/0,c_15/4,c_16/0,c_17/2,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq#,if1#,if2#,le#,min#,minsort#,rm#} and constructors {0
,cons,false,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:eq#(s(x),s(y)) -> c_4(eq#(x,y))
-->_1 eq#(s(x),0()) -> c_3():12
-->_1 eq#(0(),s(y)) -> c_2():11
-->_1 eq#(0(),0()) -> c_1():10
-->_1 eq#(s(x),s(y)) -> c_4(eq#(x,y)):1
2:S:if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs)))
-->_1 min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y)):7
-->_1 min#(cons(x,nil())) -> c_13():15
3:S:if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs)))
-->_1 min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y)):7
-->_1 min#(cons(x,nil())) -> c_13():15
4:S:if2#(false(),x,y,xs) -> c_7(rm#(x,xs))
-->_1 rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y)):9
-->_1 rm#(x,nil()) -> c_18():18
5:S:if2#(true(),x,y,xs) -> c_8(rm#(x,xs))
-->_1 rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y)):9
-->_1 rm#(x,nil()) -> c_18():18
6:S:le#(s(x),s(y)) -> c_11(le#(x,y))
-->_1 le#(s(x),0()) -> c_10():14
-->_1 le#(0(),y) -> c_9():13
-->_1 le#(s(x),s(y)) -> c_11(le#(x,y)):6
7:S:min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y))
-->_2 le#(s(x),0()) -> c_10():14
-->_2 le#(0(),y) -> c_9():13
-->_2 le#(s(x),s(y)) -> c_11(le#(x,y)):6
-->_1 if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs))):3
-->_1 if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs))):2
8:S:minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs)))
-->_3 rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y)):9
-->_2 minsort#(nil()) -> c_16():17
-->_4 min#(cons(x,nil())) -> c_13():15
-->_1 min#(cons(x,nil())) -> c_13():15
-->_2 minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs))):8
-->_4 min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y)):7
-->_1 min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y)):7
9:S:rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y))
-->_2 eq#(s(x),0()) -> c_3():12
-->_2 eq#(0(),s(y)) -> c_2():11
-->_2 eq#(0(),0()) -> c_1():10
-->_1 if2#(true(),x,y,xs) -> c_8(rm#(x,xs)):5
-->_1 if2#(false(),x,y,xs) -> c_7(rm#(x,xs)):4
-->_2 eq#(s(x),s(y)) -> c_4(eq#(x,y)):1
10:W:eq#(0(),0()) -> c_1()
11:W:eq#(0(),s(y)) -> c_2()
12:W:eq#(s(x),0()) -> c_3()
13:W:le#(0(),y) -> c_9()
14:W:le#(s(x),0()) -> c_10()
15:W:min#(cons(x,nil())) -> c_13()
16:W:min#(nil()) -> c_14()
17:W:minsort#(nil()) -> c_16()
18:W:rm#(x,nil()) -> c_18()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
16: min#(nil()) -> c_14()
17: minsort#(nil()) -> c_16()
18: rm#(x,nil()) -> c_18()
15: min#(cons(x,nil())) -> c_13()
13: le#(0(),y) -> c_9()
14: le#(s(x),0()) -> c_10()
10: eq#(0(),0()) -> c_1()
11: eq#(0(),s(y)) -> c_2()
12: eq#(s(x),0()) -> c_3()
* Step 5: Failure MAYBE
+ Considered Problem:
- Strict DPs:
eq#(s(x),s(y)) -> c_4(eq#(x,y))
if1#(false(),x,y,xs) -> c_5(min#(cons(y,xs)))
if1#(true(),x,y,xs) -> c_6(min#(cons(x,xs)))
if2#(false(),x,y,xs) -> c_7(rm#(x,xs))
if2#(true(),x,y,xs) -> c_8(rm#(x,xs))
le#(s(x),s(y)) -> c_11(le#(x,y))
min#(cons(x,cons(y,xs))) -> c_12(if1#(le(x,y),x,y,xs),le#(x,y))
minsort#(cons(x,xs)) -> c_15(min#(cons(x,xs))
,minsort#(rm(min(cons(x,xs)),cons(x,xs)))
,rm#(min(cons(x,xs)),cons(x,xs))
,min#(cons(x,xs)))
rm#(x,cons(y,xs)) -> c_17(if2#(eq(x,y),x,y,xs),eq#(x,y))
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,xs) -> min(cons(y,xs))
if1(true(),x,y,xs) -> min(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,rm(x,xs))
if2(true(),x,y,xs) -> rm(x,xs)
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
min(cons(x,cons(y,xs))) -> if1(le(x,y),x,y,xs)
min(cons(x,nil())) -> x
rm(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
rm(x,nil()) -> nil()
- Signature:
{eq/2,if1/4,if2/4,le/2,min/1,minsort/1,rm/2,eq#/2,if1#/4,if2#/4,le#/2,min#/1,minsort#/1,rm#/2} / {0/0,cons/2
,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0
,c_14/0,c_15/4,c_16/0,c_17/2,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq#,if1#,if2#,le#,min#,minsort#,rm#} and constructors {0
,cons,false,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE