* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(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_gcd/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus,pred} and constructors {0,false,s
,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(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_gcd/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus,pred} and constructors {0,false,s
,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
le(x,y){x -> s(x),y -> s(y)} =
le(s(x),s(y)) ->^+ le(x,y)
= C[le(x,y) = le(x,y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(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_gcd/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd,if_gcd,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_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(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 1.b:2: UsableRules 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_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(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) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,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#(0(),y) -> c_1()
gcd#(s(x),0()) -> c_2()
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(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()
** Step 1.b:3: 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_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(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:
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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,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_gcd#(le(y,x),s(x),s(y)),le#(y,x))
4: if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
5: if_gcd#(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 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(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:
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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,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_gcd#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):4
-->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 if_gcd#(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_gcd#(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_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
3:S:if_gcd#(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_gcd#(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 1.b:5: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(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:
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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,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_gcd#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):4
-->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2
2:S:if_gcd#(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_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
3:S:if_gcd#(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_gcd#(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 1.b:6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,le#,minus#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
4: le#(s(x),s(y)) -> c_8(le#(x,y))
The strictly oriented rules are moved into the weak component.
*** Step 1.b:6.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,le#,minus#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_3) = {1,2},
uargs(c_4) = {1,2},
uargs(c_5) = {1,2},
uargs(c_8) = {1},
uargs(c_10) = {1}
Following symbols are considered usable:
{minus,pred,gcd#,if_gcd#,le#,minus#,pred#}
TcT has computed the following interpretation:
p(0) = 0
p(false) = 0
p(gcd) = 2
p(if_gcd) = 2 + x1*x2 + 2*x1*x3 + 2*x1^2 + x2^2 + x3
p(le) = 0
p(minus) = x1
p(pred) = x1
p(s) = 1 + x1
p(true) = 0
p(gcd#) = x1 + x1^2 + x2^2
p(if_gcd#) = 1 + x2^2 + x3^2
p(le#) = x2
p(minus#) = 2
p(pred#) = x1^2
p(c_1) = 0
p(c_2) = 0
p(c_3) = x1 + x2
p(c_4) = x1 + x2
p(c_5) = x1 + x2
p(c_6) = 0
p(c_7) = 1
p(c_8) = x1
p(c_9) = 0
p(c_10) = x1
p(c_11) = 1
Following rules are strictly oriented:
le#(s(x),s(y)) = 1 + y
> y
= c_8(le#(x,y))
Following rules are (at-least) weakly oriented:
gcd#(s(x),s(y)) = 3 + 3*x + x^2 + 2*y + y^2
>= 3 + 3*x + x^2 + 2*y + y^2
= c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) = 3 + 2*x + x^2 + 2*y + y^2
>= 3 + 2*x + x^2 + y + y^2
= c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) = 3 + 2*x + x^2 + 2*y + y^2
>= 3 + x + x^2 + 2*y + y^2
= c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) = 2
>= 2
= c_10(minus#(x,y))
minus(x,0()) = x
>= x
= x
minus(x,s(y)) = x
>= x
= pred(minus(x,y))
pred(s(x)) = 1 + x
>= x
= x
*** Step 1.b:6.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) -> c_10(minus#(x,y))
- Weak DPs:
le#(s(x),s(y)) -> c_8(le#(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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,le#,minus#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) -> c_10(minus#(x,y))
- Weak DPs:
le#(s(x),s(y)) -> c_8(le#(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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,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_gcd#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):5
-->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2
2:S:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
-->_2 minus#(x,s(y)) -> c_10(minus#(x,y)):4
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
3:S:if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_10(minus#(x,y)):4
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
4:S:minus#(x,s(y)) -> c_10(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_10(minus#(x,y)):4
5:W:le#(s(x),s(y)) -> c_8(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_8(le#(x,y)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: le#(s(x),s(y)) -> c_8(le#(x,y))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_gcd#,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_gcd#(le(y,x),s(x),s(y)),le#(y,x))
-->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2
2:S:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
-->_2 minus#(x,s(y)) -> c_10(minus#(x,y)):4
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
3:S:if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_10(minus#(x,y)):4
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
4:S:minus#(x,s(y)) -> c_10(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_10(minus#(x,y)):4
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_gcd#(le(y,x),s(x),s(y)))
*** Step 1.b:6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_4/2,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_gcd#,le#,minus#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
3: if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
4: minus#(x,s(y)) -> c_10(minus#(x,y))
Consider the set of all dependency pairs
1: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)))
2: if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
3: if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
4: minus#(x,s(y)) -> c_10(minus#(x,y))
Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2))
SPACE(?,?)on application of the dependency pairs
{2,3,4}
These cover all (indirect) predecessors of dependency pairs
{1,2,3,4}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
**** Step 1.b:6.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_4/2,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_gcd#,le#,minus#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_3) = {1},
uargs(c_4) = {1,2},
uargs(c_5) = {1,2},
uargs(c_10) = {1}
Following symbols are considered usable:
{minus,pred,gcd#,if_gcd#,le#,minus#,pred#}
TcT has computed the following interpretation:
p(0) = 1
p(false) = 0
p(gcd) = 0
p(if_gcd) = 1 + 2*x1 + 2*x1*x2 + 2*x3
p(le) = 0
p(minus) = x1
p(pred) = x1
p(s) = 1 + x1
p(true) = 0
p(gcd#) = 3 + x1 + 2*x1*x2 + x2
p(if_gcd#) = 3 + x2 + 2*x2*x3 + x3
p(le#) = x1
p(minus#) = 1 + 2*x2
p(pred#) = x1^2
p(c_1) = 0
p(c_2) = 1
p(c_3) = x1
p(c_4) = 1 + x1 + x2
p(c_5) = 1 + x1 + x2
p(c_6) = 1
p(c_7) = 1
p(c_8) = 0
p(c_9) = 0
p(c_10) = 1 + x1
p(c_11) = 0
Following rules are strictly oriented:
if_gcd#(false(),s(x),s(y)) = 7 + 3*x + 2*x*y + 3*y
> 6 + 3*x + 2*x*y + 3*y
= c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) = 7 + 3*x + 2*x*y + 3*y
> 6 + 3*x + 2*x*y + 3*y
= c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) = 3 + 2*y
> 2 + 2*y
= c_10(minus#(x,y))
Following rules are (at-least) weakly oriented:
gcd#(s(x),s(y)) = 7 + 3*x + 2*x*y + 3*y
>= 7 + 3*x + 2*x*y + 3*y
= c_3(if_gcd#(le(y,x),s(x),s(y)))
minus(x,0()) = x
>= x
= x
minus(x,s(y)) = x
>= x
= pred(minus(x,y))
pred(s(x)) = 1 + x
>= x
= x
**** Step 1.b:6.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)))
- Weak DPs:
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_4/2,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_gcd#,le#,minus#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_4/2,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_gcd#,le#,minus#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)))
-->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2
2:W:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
-->_2 minus#(x,s(y)) -> c_10(minus#(x,y)):4
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y))):1
3:W:if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_10(minus#(x,y)):4
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y))):1
4:W:minus#(x,s(y)) -> c_10(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_10(minus#(x,y)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)))
3: if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
2: if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
4: minus#(x,s(y)) -> c_10(minus#(x,y))
**** Step 1.b:6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- 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_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/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_4/2,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_gcd#,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(Omega(n^1),O(n^2))