*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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
Weak DP Rules:
Weak TRS Rules:
Signature:
{gcd/2,if/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {gcd,if,le,minus,pred}/{0,false,s,true}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
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.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
UsableRules
Proof:
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#(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()
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
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()
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
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 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
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
SimplifyRHS
Proof:
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))
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
5: minus#(X,s(Y)) -> c_10(minus#(X
,Y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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#,le#,minus#,pred#}
TcT has computed the following interpretation:
p(0) = 0
p(false) = 0
p(gcd) = 2*x1^2 + 2*x2^2
p(if) = 2*x1^2
p(le) = x1*x2 + 2*x2 + x2^2
p(minus) = x1
p(pred) = x1
p(s) = 1 + x1
p(true) = 0
p(gcd#) = 3 + 2*x1*x2
p(if#) = 1 + 2*x2*x3
p(le#) = 2
p(minus#) = x2
p(pred#) = 2
p(c_1) = 1
p(c_2) = 0
p(c_3) = x1 + x2
p(c_4) = x1 + x2
p(c_5) = x1 + x2
p(c_6) = 1
p(c_7) = 0
p(c_8) = x1
p(c_9) = 0
p(c_10) = x1
p(c_11) = 1
Following rules are strictly oriented:
minus#(X,s(Y)) = 1 + Y
> Y
= c_10(minus#(X,Y))
Following rules are (at-least) weakly oriented:
gcd#(s(X),s(Y)) = 5 + 2*X + 2*X*Y + 2*Y
>= 5 + 2*X + 2*X*Y + 2*Y
= c_3(if#(le(Y,X),s(X),s(Y))
,le#(Y,X))
if#(false(),s(X),s(Y)) = 3 + 2*X + 2*X*Y + 2*Y
>= 3 + X + 2*X*Y + 2*Y
= c_4(gcd#(minus(Y,X),s(X))
,minus#(Y,X))
if#(true(),s(X),s(Y)) = 3 + 2*X + 2*X*Y + 2*Y
>= 3 + 2*X + 2*X*Y + Y
= c_5(gcd#(minus(X,Y),s(Y))
,minus#(X,Y))
le#(s(X),s(Y)) = 2
>= 2
= c_8(le#(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
*** 1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
minus#(X,s(Y)) -> c_10(minus#(X,Y))
Weak TRS 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
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
minus#(X,s(Y)) -> c_10(minus#(X,Y))
Weak TRS 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
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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(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(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:W:minus#(X,s(Y)) -> c_10(minus#(X,Y))
-->_1 minus#(X,s(Y)) -> c_10(minus#(X,Y)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: minus#(X,s(Y)) -> c_10(minus#(X
,Y))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
SimplifyRHS
Proof:
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))
-->_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
4:S:le#(s(X),s(Y)) -> c_8(le#(X,Y))
-->_1 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
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)))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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)))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
Obligation:
Innermost
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: if#(false(),s(X),s(Y)) ->
c_4(gcd#(minus(Y,X),s(X)))
3: if#(true(),s(X),s(Y)) ->
c_5(gcd#(minus(X,Y),s(Y)))
4: le#(s(X),s(Y)) -> c_8(le#(X,Y))
Consider the set of all dependency pairs
1: gcd#(s(X),s(Y)) -> c_3(if#(le(Y
,X)
,s(X)
,s(Y))
,le#(Y,X))
2: if#(false(),s(X),s(Y)) ->
c_4(gcd#(minus(Y,X),s(X)))
3: if#(true(),s(X),s(Y)) ->
c_5(gcd#(minus(X,Y),s(Y)))
4: le#(s(X),s(Y)) -> c_8(le#(X,Y))
Processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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)))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
Obligation:
Innermost
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_3) = {1,2},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1}
Following symbols are considered usable:
{minus,pred,gcd#,if#,le#,minus#,pred#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(false) = [0]
[0]
[0]
p(gcd) = [0]
[0]
[0]
p(if) = [0]
[0]
[0]
p(le) = [0 0 0] [0]
[0 0 0] x2 + [0]
[0 0 1] [0]
p(minus) = [1 0 0] [0]
[1 1 0] x1 + [0]
[0 1 1] [0]
p(pred) = [1 0 0] [0]
[1 0 0] x1 + [0]
[0 0 1] [0]
p(s) = [1 1 1] [0]
[0 0 0] x1 + [1]
[0 0 1] [1]
p(true) = [0]
[0]
[0]
p(gcd#) = [1 0 1] [1 0 0] [0]
[0 0 0] x1 + [0 1 1] x2 + [0]
[0 0 0] [0 1 0] [1]
p(if#) = [1 0 0] [1 1 0] [0]
[0 0 0] x2 + [0 1 0] x3 + [1]
[1 0 0] [1 0 0] [0]
p(le#) = [0 0 0] [0 0 1] [0]
[0 0 1] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [1]
p(minus#) = [0]
[0]
[0]
p(pred#) = [0]
[0]
[0]
p(c_1) = [0]
[0]
[0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
p(c_4) = [1 0 0] [0]
[0 0 1] x1 + [0]
[1 0 0] [0]
p(c_5) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_6) = [0]
[0]
[0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [1 0 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [0]
p(c_9) = [0]
[0]
[0]
p(c_10) = [0]
[0]
[0]
p(c_11) = [0]
[0]
[0]
Following rules are strictly oriented:
if#(false(),s(X),s(Y)) = [1 1 1] [1 1 1] [1]
[0 0 0] X + [0 0 0] Y + [2]
[1 1 1] [1 1 1] [0]
> [1 1 1] [1 1 1] [0]
[0 0 0] X + [0 0 0] Y + [2]
[1 1 1] [1 1 1] [0]
= c_4(gcd#(minus(Y,X),s(X)))
if#(true(),s(X),s(Y)) = [1 1 1] [1 1 1] [1]
[0 0 0] X + [0 0 0] Y + [2]
[1 1 1] [1 1 1] [0]
> [1 1 1] [1 1 1] [0]
[0 0 0] X + [0 0 0] Y + [0]
[0 0 0] [0 0 0] [0]
= c_5(gcd#(minus(X,Y),s(Y)))
le#(s(X),s(Y)) = [0 0 0] [0 0 1] [1]
[0 0 1] X + [0 0 0] Y + [2]
[0 0 0] [0 0 0] [1]
> [0 0 1] [0]
[0 0 0] Y + [1]
[0 0 0] [0]
= c_8(le#(X,Y))
Following rules are (at-least) weakly oriented:
gcd#(s(X),s(Y)) = [1 1 2] [1 1 1] [1]
[0 0 0] X + [0 0 1] Y + [2]
[0 0 0] [0 0 0] [2]
>= [1 1 2] [1 1 1] [1]
[0 0 0] X + [0 0 0] Y + [2]
[0 0 0] [0 0 0] [0]
= c_3(if#(le(Y,X),s(X),s(Y))
,le#(Y,X))
minus(X,0()) = [1 0 0] [0]
[1 1 0] X + [0]
[0 1 1] [0]
>= [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= X
minus(X,s(Y)) = [1 0 0] [0]
[1 1 0] X + [0]
[0 1 1] [0]
>= [1 0 0] [0]
[1 0 0] X + [0]
[0 1 1] [0]
= pred(minus(X,Y))
pred(s(X)) = [1 1 1] [0]
[1 1 1] X + [0]
[0 0 1] [1]
>= [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= X
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
Strict TRS Rules:
Weak DP Rules:
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)))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
Weak TRS 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
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_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
Obligation:
Innermost
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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)))
if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)))
le#(s(X),s(Y)) -> c_8(le#(X,Y))
Weak TRS 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
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_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
Obligation:
Innermost
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W: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))):3
-->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X))):2
2:W:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)))
-->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
3:W:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)))
-->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
4:W:le#(s(X),s(Y)) -> c_8(le#(X,Y))
-->_1 le#(s(X),s(Y)) -> c_8(le#(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#(le(Y
,X)
,s(X)
,s(Y))
,le#(Y,X))
3: if#(true(),s(X),s(Y)) ->
c_5(gcd#(minus(X,Y),s(Y)))
2: if#(false(),s(X),s(Y)) ->
c_4(gcd#(minus(Y,X),s(X)))
4: le#(s(X),s(Y)) -> c_8(le#(X,Y))
*** 1.1.1.1.1.1.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
Obligation:
Innermost
basic terms: {gcd#,if#,le#,minus#,pred#}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).