MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
gcd(x,0()) -> x
gcd(0(),s(y)) -> s(y)
gcd(s(x),s(y)) -> gcd(mod(s(x),s(y)),mod(s(y),s(x)))
if(false(),x,y) -> mod(minus(x,y),y)
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),x) -> 0()
minus(s(x),0()) -> s(x)
minus(s(x),s(y)) -> minus(x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if(lt(x,s(y)),x,s(y))
- Signature:
{gcd/2,if/3,lt/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd,if,lt,minus,mod} and constructors {0,false,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
gcd#(x,0()) -> c_1()
gcd#(0(),s(y)) -> c_2()
gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
if#(true(),x,y) -> c_5()
lt#(x,0()) -> c_6()
lt#(0(),s(x)) -> c_7()
lt#(s(x),s(y)) -> c_8(lt#(x,y))
minus#(0(),x) -> c_9()
minus#(s(x),0()) -> c_10()
minus#(s(x),s(y)) -> c_11(minus#(x,y))
mod#(x,0()) -> c_12()
mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
gcd#(x,0()) -> c_1()
gcd#(0(),s(y)) -> c_2()
gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
if#(true(),x,y) -> c_5()
lt#(x,0()) -> c_6()
lt#(0(),s(x)) -> c_7()
lt#(s(x),s(y)) -> c_8(lt#(x,y))
minus#(0(),x) -> c_9()
minus#(s(x),0()) -> c_10()
minus#(s(x),s(y)) -> c_11(minus#(x,y))
mod#(x,0()) -> c_12()
mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
- Weak TRS:
gcd(x,0()) -> x
gcd(0(),s(y)) -> s(y)
gcd(s(x),s(y)) -> gcd(mod(s(x),s(y)),mod(s(y),s(x)))
if(false(),x,y) -> mod(minus(x,y),y)
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),x) -> 0()
minus(s(x),0()) -> s(x)
minus(s(x),s(y)) -> minus(x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if(lt(x,s(y)),x,s(y))
- Signature:
{gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s
,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
if(false(),x,y) -> mod(minus(x,y),y)
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),x) -> 0()
minus(s(x),0()) -> s(x)
minus(s(x),s(y)) -> minus(x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if(lt(x,s(y)),x,s(y))
gcd#(x,0()) -> c_1()
gcd#(0(),s(y)) -> c_2()
gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
if#(true(),x,y) -> c_5()
lt#(x,0()) -> c_6()
lt#(0(),s(x)) -> c_7()
lt#(s(x),s(y)) -> c_8(lt#(x,y))
minus#(0(),x) -> c_9()
minus#(s(x),0()) -> c_10()
minus#(s(x),s(y)) -> c_11(minus#(x,y))
mod#(x,0()) -> c_12()
mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
gcd#(x,0()) -> c_1()
gcd#(0(),s(y)) -> c_2()
gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
if#(true(),x,y) -> c_5()
lt#(x,0()) -> c_6()
lt#(0(),s(x)) -> c_7()
lt#(s(x),s(y)) -> c_8(lt#(x,y))
minus#(0(),x) -> c_9()
minus#(s(x),0()) -> c_10()
minus#(s(x),s(y)) -> c_11(minus#(x,y))
mod#(x,0()) -> c_12()
mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
- Weak TRS:
if(false(),x,y) -> mod(minus(x,y),y)
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),x) -> 0()
minus(s(x),0()) -> s(x)
minus(s(x),s(y)) -> minus(x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if(lt(x,s(y)),x,s(y))
- Signature:
{gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s
,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,5,6,7,9,10,12}
by application of
Pre({1,2,5,6,7,9,10,12}) = {3,4,8,11,13}.
Here rules are labelled as follows:
1: gcd#(x,0()) -> c_1()
2: gcd#(0(),s(y)) -> c_2()
3: gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
4: if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
5: if#(true(),x,y) -> c_5()
6: lt#(x,0()) -> c_6()
7: lt#(0(),s(x)) -> c_7()
8: lt#(s(x),s(y)) -> c_8(lt#(x,y))
9: minus#(0(),x) -> c_9()
10: minus#(s(x),0()) -> c_10()
11: minus#(s(x),s(y)) -> c_11(minus#(x,y))
12: mod#(x,0()) -> c_12()
13: mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
lt#(s(x),s(y)) -> c_8(lt#(x,y))
minus#(s(x),s(y)) -> c_11(minus#(x,y))
mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
- Weak DPs:
gcd#(x,0()) -> c_1()
gcd#(0(),s(y)) -> c_2()
if#(true(),x,y) -> c_5()
lt#(x,0()) -> c_6()
lt#(0(),s(x)) -> c_7()
minus#(0(),x) -> c_9()
minus#(s(x),0()) -> c_10()
mod#(x,0()) -> c_12()
- Weak TRS:
if(false(),x,y) -> mod(minus(x,y),y)
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),x) -> 0()
minus(s(x),0()) -> s(x)
minus(s(x),s(y)) -> minus(x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if(lt(x,s(y)),x,s(y))
- Signature:
{gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s
,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
-->_3 mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))):5
-->_2 mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))):5
-->_1 gcd#(0(),s(y)) -> c_2():7
-->_1 gcd#(x,0()) -> c_1():6
-->_1 gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))):1
2:S:if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
-->_1 mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))):5
-->_2 minus#(s(x),s(y)) -> c_11(minus#(x,y)):4
-->_1 mod#(x,0()) -> c_12():13
-->_2 minus#(s(x),0()) -> c_10():12
-->_2 minus#(0(),x) -> c_9():11
3:S:lt#(s(x),s(y)) -> c_8(lt#(x,y))
-->_1 lt#(0(),s(x)) -> c_7():10
-->_1 lt#(x,0()) -> c_6():9
-->_1 lt#(s(x),s(y)) -> c_8(lt#(x,y)):3
4:S:minus#(s(x),s(y)) -> c_11(minus#(x,y))
-->_1 minus#(s(x),0()) -> c_10():12
-->_1 minus#(0(),x) -> c_9():11
-->_1 minus#(s(x),s(y)) -> c_11(minus#(x,y)):4
5:S:mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
-->_2 lt#(0(),s(x)) -> c_7():10
-->_1 if#(true(),x,y) -> c_5():8
-->_2 lt#(s(x),s(y)) -> c_8(lt#(x,y)):3
-->_1 if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)):2
6:W:gcd#(x,0()) -> c_1()
7:W:gcd#(0(),s(y)) -> c_2()
8:W:if#(true(),x,y) -> c_5()
9:W:lt#(x,0()) -> c_6()
10:W:lt#(0(),s(x)) -> c_7()
11:W:minus#(0(),x) -> c_9()
12:W:minus#(s(x),0()) -> c_10()
13:W:mod#(x,0()) -> c_12()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: gcd#(x,0()) -> c_1()
7: gcd#(0(),s(y)) -> c_2()
13: mod#(x,0()) -> c_12()
11: minus#(0(),x) -> c_9()
12: minus#(s(x),0()) -> c_10()
9: lt#(x,0()) -> c_6()
8: if#(true(),x,y) -> c_5()
10: lt#(0(),s(x)) -> c_7()
* Step 5: WeightGap MAYBE
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
lt#(s(x),s(y)) -> c_8(lt#(x,y))
minus#(s(x),s(y)) -> c_11(minus#(x,y))
mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
- Weak TRS:
if(false(),x,y) -> mod(minus(x,y),y)
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),x) -> 0()
minus(s(x),0()) -> s(x)
minus(s(x),s(y)) -> minus(x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if(lt(x,s(y)),x,s(y))
- Signature:
{gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s
,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
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(if) = {1},
uargs(mod) = {1},
uargs(gcd#) = {1,2},
uargs(if#) = {1},
uargs(mod#) = {1},
uargs(c_3) = {1,2,3},
uargs(c_4) = {1,2},
uargs(c_8) = {1},
uargs(c_11) = {1},
uargs(c_13) = {1,2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(gcd) = [0]
p(if) = [1] x1 + [1] x2 + [3] x3 + [0]
p(lt) = [0]
p(minus) = [0]
p(mod) = [1] x1 + [0]
p(s) = [0]
p(true) = [0]
p(gcd#) = [1] x1 + [1] x2 + [0]
p(if#) = [1] x1 + [1] x3 + [0]
p(lt#) = [0]
p(minus#) = [1]
p(mod#) = [1] x1 + [1] x2 + [6]
p(c_1) = [0]
p(c_2) = [2]
p(c_3) = [1] x1 + [1] x2 + [1] x3 + [2]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [1]
p(c_8) = [1] x1 + [2]
p(c_9) = [1]
p(c_10) = [0]
p(c_11) = [1] x1 + [2]
p(c_12) = [4]
p(c_13) = [1] x1 + [1] x2 + [4]
Following rules are strictly oriented:
mod#(x,s(y)) = [1] x + [6]
> [4]
= c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
Following rules are (at-least) weakly oriented:
gcd#(s(x),s(y)) = [0]
>= [14]
= c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
if#(false(),x,y) = [1] y + [0]
>= [1] y + [7]
= c_4(mod#(minus(x,y),y),minus#(x,y))
lt#(s(x),s(y)) = [0]
>= [2]
= c_8(lt#(x,y))
minus#(s(x),s(y)) = [1]
>= [3]
= c_11(minus#(x,y))
if(false(),x,y) = [1] x + [3] y + [0]
>= [0]
= mod(minus(x,y),y)
if(true(),x,y) = [1] x + [3] y + [0]
>= [1] x + [0]
= x
lt(x,0()) = [0]
>= [0]
= false()
lt(0(),s(x)) = [0]
>= [0]
= true()
lt(s(x),s(y)) = [0]
>= [0]
= lt(x,y)
minus(0(),x) = [0]
>= [0]
= 0()
minus(s(x),0()) = [0]
>= [0]
= s(x)
minus(s(x),s(y)) = [0]
>= [0]
= minus(x,y)
mod(x,0()) = [1] x + [0]
>= [0]
= 0()
mod(x,s(y)) = [1] x + [0]
>= [1] x + [0]
= if(lt(x,s(y)),x,s(y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: Failure MAYBE
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x)))
if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y))
lt#(s(x),s(y)) -> c_8(lt#(x,y))
minus#(s(x),s(y)) -> c_11(minus#(x,y))
- Weak DPs:
mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y)))
- Weak TRS:
if(false(),x,y) -> mod(minus(x,y),y)
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),x) -> 0()
minus(s(x),0()) -> s(x)
minus(s(x),s(y)) -> minus(x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if(lt(x,s(y)),x,s(y))
- Signature:
{gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s
,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE