*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
purge(nil()) -> nil()
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Weak DP Rules:
Weak TRS Rules:
Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
Obligation:
Innermost
basic terms: {eq,ifrm,purge,rm}/{0,add,false,nil,s,true}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
eq#(0(),0()) -> c_1()
eq#(0(),s(X)) -> c_2()
eq#(s(X),0()) -> c_3()
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
purge#(nil()) -> c_8()
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
rm#(N,nil()) -> c_10()
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
eq#(0(),0()) -> c_1()
eq#(0(),s(X)) -> c_2()
eq#(s(X),0()) -> c_3()
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
purge#(nil()) -> c_8()
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
rm#(N,nil()) -> c_10()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
purge(nil()) -> nil()
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
eq#(0(),0()) -> c_1()
eq#(0(),s(X)) -> c_2()
eq#(s(X),0()) -> c_3()
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
purge#(nil()) -> c_8()
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
rm#(N,nil()) -> c_10()
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
eq#(0(),0()) -> c_1()
eq#(0(),s(X)) -> c_2()
eq#(s(X),0()) -> c_3()
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
purge#(nil()) -> c_8()
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
rm#(N,nil()) -> c_10()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,2,3,8,10}
by application of
Pre({1,2,3,8,10}) = {4,5,6,7,9}.
Here rules are labelled as follows:
1: eq#(0(),0()) -> c_1()
2: eq#(0(),s(X)) -> c_2()
3: eq#(s(X),0()) -> c_3()
4: eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
5: ifrm#(false(),N,add(M,X)) ->
c_5(rm#(N,X))
6: ifrm#(true(),N,add(M,X)) ->
c_6(rm#(N,X))
7: purge#(add(N,X)) ->
c_7(purge#(rm(N,X)),rm#(N,X))
8: purge#(nil()) -> c_8()
9: rm#(N,add(M,X)) ->
c_9(ifrm#(eq(N,M),N,add(M,X))
,eq#(N,M))
10: rm#(N,nil()) -> c_10()
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Strict TRS Rules:
Weak DP Rules:
eq#(0(),0()) -> c_1()
eq#(0(),s(X)) -> c_2()
eq#(s(X),0()) -> c_3()
purge#(nil()) -> c_8()
rm#(N,nil()) -> c_10()
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
-->_1 eq#(s(X),0()) -> c_3():8
-->_1 eq#(0(),s(X)) -> c_2():7
-->_1 eq#(0(),0()) -> c_1():6
-->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1
2:S:ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):5
-->_1 rm#(N,nil()) -> c_10():10
3:S:ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):5
-->_1 rm#(N,nil()) -> c_10():10
4:S:purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
-->_2 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):5
-->_2 rm#(N,nil()) -> c_10():10
-->_1 purge#(nil()) -> c_8():9
-->_1 purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X)):4
5:S:rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
-->_2 eq#(s(X),0()) -> c_3():8
-->_2 eq#(0(),s(X)) -> c_2():7
-->_2 eq#(0(),0()) -> c_1():6
-->_1 ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X)):3
-->_1 ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X)):2
-->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1
6:W:eq#(0(),0()) -> c_1()
7:W:eq#(0(),s(X)) -> c_2()
8:W:eq#(s(X),0()) -> c_3()
9:W:purge#(nil()) -> c_8()
10:W:rm#(N,nil()) -> c_10()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: purge#(nil()) -> c_8()
10: rm#(N,nil()) -> c_10()
6: eq#(0(),0()) -> c_1()
7: eq#(0(),s(X)) -> c_2()
8: eq#(s(X),0()) -> c_3()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
Strict TRS Rules:
Weak DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Problem (S)
Strict DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Strict TRS Rules:
Weak DP Rules:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
Strict TRS Rules:
Weak DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,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:
1: eq#(s(X),s(Y)) -> c_4(eq#(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:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
Strict TRS Rules:
Weak DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,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_4) = {1},
uargs(c_5) = {1},
uargs(c_6) = {1},
uargs(c_7) = {1,2},
uargs(c_9) = {1,2}
Following symbols are considered usable:
{ifrm,rm,eq#,ifrm#,purge#,rm#}
TcT has computed the following interpretation:
p(0) = 0
p(add) = 1 + x1 + x2
p(eq) = 0
p(false) = 0
p(ifrm) = x3
p(nil) = 0
p(purge) = x1^2
p(rm) = x2
p(s) = 1 + x1
p(true) = 0
p(eq#) = x1
p(ifrm#) = 1 + x2*x3 + x3
p(purge#) = 2 + 2*x1^2
p(rm#) = 2 + x1 + x1*x2 + x2
p(c_1) = 0
p(c_2) = 0
p(c_3) = 0
p(c_4) = x1
p(c_5) = x1
p(c_6) = x1
p(c_7) = x1 + x2
p(c_8) = 0
p(c_9) = 1 + x1 + x2
p(c_10) = 0
Following rules are strictly oriented:
eq#(s(X),s(Y)) = 1 + X
> X
= c_4(eq#(X,Y))
Following rules are (at-least) weakly oriented:
ifrm#(false(),N,add(M,X)) = 2 + M + M*N + N + N*X + X
>= 2 + N + N*X + X
= c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) = 2 + M + M*N + N + N*X + X
>= 2 + N + N*X + X
= c_6(rm#(N,X))
purge#(add(N,X)) = 4 + 4*N + 4*N*X + 2*N^2 + 4*X + 2*X^2
>= 4 + N + N*X + X + 2*X^2
= c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) = 3 + M + M*N + 2*N + N*X + X
>= 3 + M + M*N + 2*N + N*X + X
= c_9(ifrm#(eq(N,M),N,add(M,X))
,eq#(N,M))
ifrm(false(),N,add(M,X)) = 1 + M + X
>= 1 + M + X
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = 1 + M + X
>= X
= rm(N,X)
rm(N,add(M,X)) = 1 + M + X
>= 1 + M + X
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = 0
>= 0
= nil()
*** 1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
-->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1
2:W:ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):5
3:W:ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):5
4:W:purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
-->_2 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):5
-->_1 purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X)):4
5:W:rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
-->_1 ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X)):3
-->_1 ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X)):2
-->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: purge#(add(N,X)) ->
c_7(purge#(rm(N,X)),rm#(N,X))
2: ifrm#(false(),N,add(M,X)) ->
c_5(rm#(N,X))
5: rm#(N,add(M,X)) ->
c_9(ifrm#(eq(N,M),N,add(M,X))
,eq#(N,M))
3: ifrm#(true(),N,add(M,X)) ->
c_6(rm#(N,X))
1: eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
*** 1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Strict TRS Rules:
Weak DP Rules:
eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):4
2:S:ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):4
3:S:purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
-->_2 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):4
-->_1 purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X)):3
4:S:rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
-->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):5
-->_1 ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X)):2
-->_1 ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X)):1
5:W:eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
-->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: eq#(s(X),s(Y)) -> c_4(eq#(X,Y))
*** 1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):4
2:S:ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):4
3:S:purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
-->_2 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M)):4
-->_1 purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X)):3
4:S:rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)),eq#(N,M))
-->_1 ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X)):2
-->_1 ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
*** 1.1.1.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
Strict TRS Rules:
Weak DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Problem (S)
Strict DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
Strict TRS Rules:
Weak DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
*** 1.1.1.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
Strict TRS Rules:
Weak DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,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:
4: rm#(N,add(M,X)) ->
c_9(ifrm#(eq(N,M),N,add(M,X)))
Consider the set of all dependency pairs
1: ifrm#(false(),N,add(M,X)) ->
c_5(rm#(N,X))
2: ifrm#(true(),N,add(M,X)) ->
c_6(rm#(N,X))
3: purge#(add(N,X)) ->
c_7(purge#(rm(N,X)),rm#(N,X))
4: rm#(N,add(M,X)) ->
c_9(ifrm#(eq(N,M),N,add(M,X)))
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
{4}
These cover all (indirect) predecessors of dependency pairs
{1,2,4}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
Strict TRS Rules:
Weak DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,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_5) = {1},
uargs(c_6) = {1},
uargs(c_7) = {1,2},
uargs(c_9) = {1}
Following symbols are considered usable:
{ifrm,rm,eq#,ifrm#,purge#,rm#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(add) = [0 0 0] [0 0 1] [0]
[0 0 1] x1 + [0 1 1] x2 + [0]
[0 0 0] [0 0 1] [1]
p(eq) = [0]
[0]
[1]
p(false) = [0]
[0]
[0]
p(ifrm) = [0 1 1] [1 0 0] [1]
[0 0 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [1 0 0] [1]
p(nil) = [0]
[0]
[0]
p(purge) = [0]
[0]
[0]
p(rm) = [0 1 1] [0 0 1] [0]
[0 0 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 1] [0]
p(s) = [0]
[0]
[0]
p(true) = [0]
[0]
[0]
p(eq#) = [0]
[0]
[0]
p(ifrm#) = [0 0 1] [1 0 0] [0]
[1 0 0] x2 + [0 1 1] x3 + [0]
[1 0 0] [1 0 0] [0]
p(purge#) = [0 1 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [1]
p(rm#) = [0 0 1] [0 0 1] [0]
[1 1 1] x1 + [0 1 1] x2 + [0]
[1 0 0] [0 1 0] [0]
p(c_1) = [0]
[0]
[0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [0]
[0]
[0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[0 0 1] x1 + [1]
[0 0 0] [0]
p(c_6) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_7) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [1]
[0 0 0] [0 0 0] [1]
p(c_8) = [0]
[0]
[0]
p(c_9) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(c_10) = [0]
[0]
[0]
Following rules are strictly oriented:
rm#(N,add(M,X)) = [0 0 0] [0 0 1] [0 0
1] [1]
[0 0 1] M + [1 1 1] N + [0 1
2] X + [1]
[0 0 1] [1 0 0] [0 1
1] [0]
> [0 0 0] [0 0 1] [0 0
1] [0]
[0 0 1] M + [1 0 0] N + [0 1
2] X + [1]
[0 0 0] [0 0 0] [0 0
0] [0]
= c_9(ifrm#(eq(N,M),N,add(M,X)))
Following rules are (at-least) weakly oriented:
ifrm#(false(),N,add(M,X)) = [0 0 0] [0 0 1] [0 0
1] [0]
[0 0 1] M + [1 0 0] N + [0 1
2] X + [1]
[0 0 0] [1 0 0] [0 0
1] [0]
>= [0 0 1] [0 0 1] [0]
[1 0 0] N + [0 1 0] X + [1]
[0 0 0] [0 0 0] [0]
= c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) = [0 0 0] [0 0 1] [0 0
1] [0]
[0 0 1] M + [1 0 0] N + [0 1
2] X + [1]
[0 0 0] [1 0 0] [0 0
1] [0]
>= [0 0 1] [0 0 1] [0]
[0 0 0] N + [0 0 0] X + [0]
[0 0 0] [0 0 0] [0]
= c_6(rm#(N,X))
purge#(add(N,X)) = [0 0 1] [0 1 1] [0]
[0 0 0] N + [0 0 0] X + [1]
[0 0 0] [0 0 0] [1]
>= [0 0 1] [0 1 1] [0]
[0 0 0] N + [0 0 0] X + [1]
[0 0 0] [0 0 0] [1]
= c_7(purge#(rm(N,X)),rm#(N,X))
ifrm(false(),N,add(M,X)) = [0 0 0] [0 1 1] [0 0
1] [1]
[0 0 1] M + [0 0 0] N + [0 1
1] X + [0]
[0 0 0] [0 0 0] [0 0
1] [1]
>= [0 0 0] [0 0 1] [0]
[0 0 1] M + [0 1 1] X + [0]
[0 0 0] [0 0 1] [1]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [0 0 0] [0 1 1] [0 0
1] [1]
[0 0 1] M + [0 0 0] N + [0 1
1] X + [0]
[0 0 0] [0 0 0] [0 0
1] [1]
>= [0 1 1] [0 0 1] [0]
[0 0 0] N + [0 1 0] X + [0]
[0 0 0] [0 0 1] [0]
= rm(N,X)
rm(N,add(M,X)) = [0 0 0] [0 1 1] [0 0
1] [1]
[0 0 1] M + [0 0 0] N + [0 1
1] X + [0]
[0 0 0] [0 0 0] [0 0
1] [1]
>= [0 0 0] [0 1 1] [0 0
1] [1]
[0 0 1] M + [0 0 0] N + [0 1
1] X + [0]
[0 0 0] [0 0 0] [0 0
1] [1]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [0 1 1] [0]
[0 0 0] N + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= nil()
*** 1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
Strict TRS Rules:
Weak DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X))):4
2:W:ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X))):4
3:W:purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
-->_2 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X))):4
-->_1 purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X)):3
4:W:rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
-->_1 ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X)):2
-->_1 ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: purge#(add(N,X)) ->
c_7(purge#(rm(N,X)),rm#(N,X))
1: ifrm#(false(),N,add(M,X)) ->
c_5(rm#(N,X))
4: rm#(N,add(M,X)) ->
c_9(ifrm#(eq(N,M),N,add(M,X)))
2: ifrm#(true(),N,add(M,X)) ->
c_6(rm#(N,X))
*** 1.1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
Strict TRS Rules:
Weak DP Rules:
ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
-->_2 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X))):4
-->_1 purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X)):1
2:W:ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X))):4
3:W:ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X))
-->_1 rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X))):4
4:W:rm#(N,add(M,X)) -> c_9(ifrm#(eq(N,M),N,add(M,X)))
-->_1 ifrm#(true(),N,add(M,X)) -> c_6(rm#(N,X)):3
-->_1 ifrm#(false(),N,add(M,X)) -> c_5(rm#(N,X)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: rm#(N,add(M,X)) ->
c_9(ifrm#(eq(N,M),N,add(M,X)))
3: ifrm#(true(),N,add(M,X)) ->
c_6(rm#(N,X))
2: ifrm#(false(),N,add(M,X)) ->
c_5(rm#(N,X))
*** 1.1.1.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_7/2,c_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X))
-->_1 purge#(add(N,X)) -> c_7(purge#(rm(N,X)),rm#(N,X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)))
*** 1.1.1.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: purge#(add(N,X)) ->
c_7(purge#(rm(N,X)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, 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:
The following argument positions are considered usable:
uargs(c_7) = {1}
Following symbols are considered usable:
{ifrm,rm,eq#,ifrm#,purge#,rm#}
TcT has computed the following interpretation:
p(0) = [2]
p(add) = [1] x2 + [1]
p(eq) = [0]
p(false) = [0]
p(ifrm) = [1] x3 + [0]
p(nil) = [3]
p(purge) = [0]
p(rm) = [1] x2 + [0]
p(s) = [1] x1 + [2]
p(true) = [0]
p(eq#) = [1] x2 + [0]
p(ifrm#) = [8] x1 + [2] x2 + [1] x3 + [1]
p(purge#) = [8] x1 + [0]
p(rm#) = [1] x1 + [1] x2 + [0]
p(c_1) = [2]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [1] x1 + [0]
p(c_6) = [1] x1 + [1]
p(c_7) = [1] x1 + [6]
p(c_8) = [2]
p(c_9) = [1]
p(c_10) = [1]
Following rules are strictly oriented:
purge#(add(N,X)) = [8] X + [8]
> [8] X + [6]
= c_7(purge#(rm(N,X)))
Following rules are (at-least) weakly oriented:
ifrm(false(),N,add(M,X)) = [1] X + [1]
>= [1] X + [1]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [1] X + [1]
>= [1] X + [0]
= rm(N,X)
rm(N,add(M,X)) = [1] X + [1]
>= [1] X + [1]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [3]
>= [3]
= nil()
*** 1.1.1.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
purge#(add(N,X)) -> c_7(purge#(rm(N,X)))
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:purge#(add(N,X)) -> c_7(purge#(rm(N,X)))
-->_1 purge#(add(N,X)) -> c_7(purge#(rm(N,X))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: purge#(add(N,X)) ->
c_7(purge#(rm(N,X)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
Signature:
{eq/2,ifrm/3,purge/1,rm/2,eq#/2,ifrm#/3,purge#/1,rm#/2} / {0/0,add/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_8/0,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {eq#,ifrm#,purge#,rm#}/{0,add,false,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).