WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
gcd(0(),Y) -> 0()
gcd(s(X),0()) -> s(X)
gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X))
if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd,if,le,minus,pred} and constructors {0,false,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
gcd#(0(),Y) -> c_1()
gcd#(s(X),0()) -> c_2()
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
le#(0(),Y) -> c_6()
le#(s(X),0()) -> c_7()
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,0()) -> c_9()
minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
pred#(s(X)) -> c_11()
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(0(),Y) -> c_1()
gcd#(s(X),0()) -> c_2()
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
le#(0(),Y) -> c_6()
le#(s(X),0()) -> c_7()
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,0()) -> c_9()
minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
pred#(s(X)) -> c_11()
- Weak TRS:
gcd(0(),Y) -> 0()
gcd(s(X),0()) -> s(X)
gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X))
if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,6,7,9,11}
by application of
Pre({1,2,6,7,9,11}) = {3,4,5,8,10}.
Here rules are labelled as follows:
1: gcd#(0(),Y) -> c_1()
2: gcd#(s(X),0()) -> c_2()
3: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
4: if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
5: if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
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: minus#(X,0()) -> c_9()
10: minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
11: pred#(s(X)) -> c_11()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
- Weak DPs:
gcd#(0(),Y) -> c_1()
gcd#(s(X),0()) -> c_2()
le#(0(),Y) -> c_6()
le#(s(X),0()) -> c_7()
minus#(X,0()) -> c_9()
pred#(s(X)) -> c_11()
- Weak TRS:
gcd(0(),Y) -> 0()
gcd(s(X),0()) -> s(X)
gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X))
if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
-->_2 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
-->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3
-->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2
-->_2 le#(s(X),0()) -> c_7():9
-->_2 le#(0(),Y) -> c_6():8
2:S:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
-->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
-->_2 minus#(X,0()) -> c_9():10
-->_1 gcd#(0(),Y) -> c_1():6
-->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
3:S:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
-->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
-->_2 minus#(X,0()) -> c_9():10
-->_1 gcd#(0(),Y) -> c_1():6
-->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
4:S:le#(s(X),s(Y)) -> c_8(le#(X,Y))
-->_1 le#(s(X),0()) -> c_7():9
-->_1 le#(0(),Y) -> c_6():8
-->_1 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
5:S:minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
-->_1 pred#(s(X)) -> c_11():11
-->_2 minus#(X,0()) -> c_9():10
-->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
6:W:gcd#(0(),Y) -> c_1()
7:W:gcd#(s(X),0()) -> c_2()
8:W:le#(0(),Y) -> c_6()
9:W:le#(s(X),0()) -> c_7()
10:W:minus#(X,0()) -> c_9()
11:W:pred#(s(X)) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: gcd#(s(X),0()) -> c_2()
6: gcd#(0(),Y) -> c_1()
10: minus#(X,0()) -> c_9()
11: pred#(s(X)) -> c_11()
8: le#(0(),Y) -> c_6()
9: le#(s(X),0()) -> c_7()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
- Weak TRS:
gcd(0(),Y) -> 0()
gcd(s(X),0()) -> s(X)
gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X))
if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
-->_2 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
-->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3
-->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2
2:S:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
-->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
-->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
3:S:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
-->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
-->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
4:S:le#(s(X),s(Y)) -> c_8(le#(X,Y))
-->_1 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
5:S:minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
-->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
minus#(X,s(Y)) -> c_10(minus#(X,Y))
* Step 5: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,s(Y)) -> c_10(minus#(X,Y))
- Weak TRS:
gcd(0(),Y) -> 0()
gcd(s(X),0()) -> s(X)
gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X))
if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,s(Y)) -> c_10(minus#(X,Y))
* Step 6: DecomposeDG WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,s(Y)) -> c_10(minus#(X,Y))
- Weak TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
and a lower component
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,s(Y)) -> c_10(minus#(X,Y))
Further, following extension rules are added to the lower component.
gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y))
gcd#(s(X),s(Y)) -> le#(Y,X)
if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X))
if#(false(),s(X),s(Y)) -> minus#(Y,X)
if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y))
if#(true(),s(X),s(Y)) -> minus#(X,Y)
** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
- Weak TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
-->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3
-->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2
2:S:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
-->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
3:S:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
-->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)))
** Step 6.a:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)))
- Weak TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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:
The following argument positions are considered usable:
uargs(pred) = {1},
uargs(gcd#) = {1},
uargs(if#) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(false) = [5]
p(gcd) = [1] x1 + [0]
p(if) = [2] x1 + [1] x2 + [1] x3 + [0]
p(le) = [7]
p(minus) = [1] x1 + [1]
p(pred) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(true) = [7]
p(gcd#) = [1] x1 + [1] x2 + [2]
p(if#) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le#) = [2]
p(minus#) = [1] x2 + [0]
p(pred#) = [1] x1 + [4]
p(c_1) = [4]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [2]
p(c_8) = [1] x1 + [1]
p(c_9) = [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
Following rules are strictly oriented:
if#(false(),s(X),s(Y)) = [1] X + [1] Y + [7]
> [1] X + [1] Y + [4]
= c_4(gcd#(minus(Y,X),s(X)))
if#(true(),s(X),s(Y)) = [1] X + [1] Y + [9]
> [1] X + [1] Y + [4]
= c_5(gcd#(minus(X,Y),s(Y)))
Following rules are (at-least) weakly oriented:
gcd#(s(X),s(Y)) = [1] X + [1] Y + [4]
>= [1] X + [1] Y + [9]
= c_3(if#(le(Y,X),s(X),s(Y)))
le(0(),Y) = [7]
>= [7]
= true()
le(s(X),0()) = [7]
>= [5]
= false()
le(s(X),s(Y)) = [7]
>= [7]
= le(X,Y)
minus(X,0()) = [1] X + [1]
>= [1] X + [0]
= X
minus(X,s(Y)) = [1] X + [1]
>= [1] X + [1]
= pred(minus(X,Y))
pred(s(X)) = [1] X + [1]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
- Weak DPs:
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)))
- Weak TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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:
The following argument positions are considered usable:
uargs(pred) = {1},
uargs(gcd#) = {1},
uargs(if#) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [1]
p(gcd) = [1] x2 + [1]
p(if) = [2] x1 + [4] x2 + [1]
p(le) = [2]
p(minus) = [1] x1 + [0]
p(pred) = [1] x1 + [0]
p(s) = [1] x1 + [2]
p(true) = [1]
p(gcd#) = [1] x1 + [1] x2 + [3]
p(if#) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le#) = [0]
p(minus#) = [1] x1 + [1] x2 + [0]
p(pred#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [2]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [0]
p(c_8) = [1] x1 + [4]
p(c_9) = [2]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
Following rules are strictly oriented:
gcd#(s(X),s(Y)) = [1] X + [1] Y + [7]
> [1] X + [1] Y + [6]
= c_3(if#(le(Y,X),s(X),s(Y)))
Following rules are (at-least) weakly oriented:
if#(false(),s(X),s(Y)) = [1] X + [1] Y + [5]
>= [1] X + [1] Y + [5]
= c_4(gcd#(minus(Y,X),s(X)))
if#(true(),s(X),s(Y)) = [1] X + [1] Y + [5]
>= [1] X + [1] Y + [5]
= c_5(gcd#(minus(X,Y),s(Y)))
le(0(),Y) = [2]
>= [1]
= true()
le(s(X),0()) = [2]
>= [1]
= false()
le(s(X),s(Y)) = [2]
>= [2]
= le(X,Y)
minus(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
minus(X,s(Y)) = [1] X + [0]
>= [1] X + [0]
= pred(minus(X,Y))
pred(s(X)) = [1] X + [2]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.a:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)))
- Weak TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 6.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,s(Y)) -> c_10(minus#(X,Y))
- Weak DPs:
gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y))
gcd#(s(X),s(Y)) -> le#(Y,X)
if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X))
if#(false(),s(X),s(Y)) -> minus#(Y,X)
if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y))
if#(true(),s(X),s(Y)) -> minus#(X,Y)
- Weak TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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:
The following argument positions are considered usable:
uargs(pred) = {1},
uargs(gcd#) = {1},
uargs(if#) = {1},
uargs(c_8) = {1},
uargs(c_10) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [1]
p(gcd) = [0]
p(if) = [2] x2 + [0]
p(le) = [1]
p(minus) = [1] x1 + [1]
p(pred) = [1] x1 + [0]
p(s) = [1] x1 + [4]
p(true) = [1]
p(gcd#) = [1] x1 + [1] x2 + [4]
p(if#) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le#) = [0]
p(minus#) = [1] x2 + [0]
p(pred#) = [1] x1 + [1]
p(c_1) = [4]
p(c_2) = [4]
p(c_3) = [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [1]
p(c_6) = [1]
p(c_7) = [2]
p(c_8) = [1] x1 + [2]
p(c_9) = [4]
p(c_10) = [1] x1 + [0]
p(c_11) = [1]
Following rules are strictly oriented:
minus#(X,s(Y)) = [1] Y + [4]
> [1] Y + [0]
= c_10(minus#(X,Y))
Following rules are (at-least) weakly oriented:
gcd#(s(X),s(Y)) = [1] X + [1] Y + [12]
>= [1] X + [1] Y + [9]
= if#(le(Y,X),s(X),s(Y))
gcd#(s(X),s(Y)) = [1] X + [1] Y + [12]
>= [0]
= le#(Y,X)
if#(false(),s(X),s(Y)) = [1] X + [1] Y + [9]
>= [1] X + [1] Y + [9]
= gcd#(minus(Y,X),s(X))
if#(false(),s(X),s(Y)) = [1] X + [1] Y + [9]
>= [1] X + [0]
= minus#(Y,X)
if#(true(),s(X),s(Y)) = [1] X + [1] Y + [9]
>= [1] X + [1] Y + [9]
= gcd#(minus(X,Y),s(Y))
if#(true(),s(X),s(Y)) = [1] X + [1] Y + [9]
>= [1] Y + [0]
= minus#(X,Y)
le#(s(X),s(Y)) = [0]
>= [2]
= c_8(le#(X,Y))
le(0(),Y) = [1]
>= [1]
= true()
le(s(X),0()) = [1]
>= [1]
= false()
le(s(X),s(Y)) = [1]
>= [1]
= le(X,Y)
minus(X,0()) = [1] X + [1]
>= [1] X + [0]
= X
minus(X,s(Y)) = [1] X + [1]
>= [1] X + [1]
= pred(minus(X,Y))
pred(s(X)) = [1] X + [4]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
le#(s(X),s(Y)) -> c_8(le#(X,Y))
- Weak DPs:
gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y))
gcd#(s(X),s(Y)) -> le#(Y,X)
if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X))
if#(false(),s(X),s(Y)) -> minus#(Y,X)
if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y))
if#(true(),s(X),s(Y)) -> minus#(X,Y)
minus#(X,s(Y)) -> c_10(minus#(X,Y))
- Weak TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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:
The following argument positions are considered usable:
uargs(pred) = {1},
uargs(gcd#) = {1},
uargs(if#) = {1},
uargs(c_8) = {1},
uargs(c_10) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(gcd) = [0]
p(if) = [0]
p(le) = [1]
p(minus) = [1] x1 + [0]
p(pred) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(true) = [0]
p(gcd#) = [1] x1 + [1] x2 + [1]
p(if#) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le#) = [1] x2 + [0]
p(minus#) = [0]
p(pred#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
Following rules are strictly oriented:
le#(s(X),s(Y)) = [1] Y + [1]
> [1] Y + [0]
= c_8(le#(X,Y))
Following rules are (at-least) weakly oriented:
gcd#(s(X),s(Y)) = [1] X + [1] Y + [3]
>= [1] X + [1] Y + [3]
= if#(le(Y,X),s(X),s(Y))
gcd#(s(X),s(Y)) = [1] X + [1] Y + [3]
>= [1] X + [0]
= le#(Y,X)
if#(false(),s(X),s(Y)) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [2]
= gcd#(minus(Y,X),s(X))
if#(false(),s(X),s(Y)) = [1] X + [1] Y + [2]
>= [0]
= minus#(Y,X)
if#(true(),s(X),s(Y)) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [2]
= gcd#(minus(X,Y),s(Y))
if#(true(),s(X),s(Y)) = [1] X + [1] Y + [2]
>= [0]
= minus#(X,Y)
minus#(X,s(Y)) = [0]
>= [0]
= c_10(minus#(X,Y))
le(0(),Y) = [1]
>= [0]
= true()
le(s(X),0()) = [1]
>= [0]
= false()
le(s(X),s(Y)) = [1]
>= [1]
= le(X,Y)
minus(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
minus(X,s(Y)) = [1] X + [0]
>= [1] X + [0]
= pred(minus(X,Y))
pred(s(X)) = [1] X + [1]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.b:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y))
gcd#(s(X),s(Y)) -> le#(Y,X)
if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X))
if#(false(),s(X),s(Y)) -> minus#(Y,X)
if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y))
if#(true(),s(X),s(Y)) -> minus#(X,Y)
le#(s(X),s(Y)) -> c_8(le#(X,Y))
minus#(X,s(Y)) -> c_10(minus#(X,Y))
- Weak TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(X,0()) -> X
minus(X,s(Y)) -> pred(minus(X,Y))
pred(s(X)) -> X
- Signature:
{gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))