We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X))
, purge(nil()) -> nil()
, purge(add(N, X)) -> add(N, purge(rm(N, X))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
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))
, rm^#(N, nil()) -> c_5()
, rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
, purge^#(nil()) -> c_9()
, purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
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))
, rm^#(N, nil()) -> c_5()
, rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
, purge^#(nil()) -> c_9()
, purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X))
, purge(nil()) -> nil()
, purge(add(N, X)) -> add(N, purge(rm(N, X))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of {1,2,3,5,9} by
applications of Pre({1,2,3,5,9}) = {4,6,7,8,10}. Here rules are
labeled as follows:
DPs:
{ 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: rm^#(N, nil()) -> c_5()
, 6: rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, 7: ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, 8: ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
, 9: purge^#(nil()) -> c_9()
, 10: purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ eq^#(s(X), s(Y)) -> c_4(eq^#(X, Y))
, rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
, purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak DPs:
{ eq^#(0(), 0()) -> c_1()
, eq^#(0(), s(X)) -> c_2()
, eq^#(s(X), 0()) -> c_3()
, rm^#(N, nil()) -> c_5()
, purge^#(nil()) -> c_9() }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X))
, purge(nil()) -> nil()
, purge(add(N, X)) -> add(N, purge(rm(N, X))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ eq^#(0(), 0()) -> c_1()
, eq^#(0(), s(X)) -> c_2()
, eq^#(s(X), 0()) -> c_3()
, rm^#(N, nil()) -> c_5()
, purge^#(nil()) -> c_9() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ eq^#(s(X), s(Y)) -> c_4(eq^#(X, Y))
, rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
, purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X))
, purge(nil()) -> nil()
, purge(add(N, X)) -> add(N, purge(rm(N, X))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We replace rewrite rules by usable rules:
Weak Usable Rules:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ eq^#(s(X), s(Y)) -> c_4(eq^#(X, Y))
, rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
, purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We decompose the input problem according to the dependency graph
into the upper component
{ purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
and lower component
{ eq^#(s(X), s(Y)) -> c_4(eq^#(X, Y))
, rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X)) }
Further, following extension rules are added to the lower
component.
{ purge^#(add(N, X)) -> rm^#(N, X)
, purge^#(add(N, X)) -> purge^#(rm(N, X)) }
TcT solves the upper component with certificate YES(O(1),O(n^1)).
Sub-proof:
----------
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.
DPs:
{ 1: purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Trs: { ifrm(true(), N, add(M, X)) -> rm(N, X) }
Sub-proof:
----------
The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping
safe(eq) = {1, 2}, safe(0) = {}, safe(true) = {}, safe(s) = {1},
safe(false) = {}, safe(rm) = {2}, safe(nil) = {},
safe(add) = {1, 2}, safe(ifrm) = {1, 3}, safe(rm^#) = {},
safe(purge^#) = {}, safe(c_10) = {}
and precedence
empty .
Following symbols are considered recursive:
{purge^#}
The recursion depth is 1.
Further, following argument filtering is employed:
pi(eq) = [1, 2], pi(0) = [], pi(true) = [], pi(s) = [],
pi(false) = [], pi(rm) = 2, pi(nil) = [], pi(add) = [2],
pi(ifrm) = 3, pi(rm^#) = [], pi(purge^#) = [1], pi(c_10) = [1, 2]
Usable defined function symbols are a subset of:
{rm, ifrm, rm^#, purge^#}
For your convenience, here are the satisfied ordering constraints:
pi(purge^#(add(N, X))) = purge^#(add(; X);)
> c_10(purge^#(X;), rm^#();)
= pi(c_10(purge^#(rm(N, X)), rm^#(N, X)))
pi(rm(N, nil())) = nil()
>= nil()
= pi(nil())
pi(rm(N, add(M, X))) = add(; X)
>= add(; X)
= pi(ifrm(eq(N, M), N, add(M, X)))
pi(ifrm(true(), N, add(M, X))) = add(; X)
> X
= pi(rm(N, X))
pi(ifrm(false(), N, add(M, X))) = add(; X)
>= add(; X)
= pi(add(M, rm(N, X)))
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs:
{ purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
No rule is usable, rules are removed from the input problem.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ eq^#(s(X), s(Y)) -> c_4(eq^#(X, Y))
, rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X)) }
Weak DPs:
{ purge^#(add(N, X)) -> rm^#(N, X)
, purge^#(add(N, X)) -> purge^#(rm(N, X)) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 2' to
orient following rules strictly.
DPs:
{ 1: eq^#(s(X), s(Y)) -> c_4(eq^#(X, Y)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_4) = {1}, Uargs(c_6) = {1, 2}, Uargs(c_7) = {1},
Uargs(c_8) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).
[eq](x1, x2) = [0]
[0]
[0] = [0]
[0]
[true] = [0]
[0]
[s](x1) = [1 0] x1 + [4]
[0 0] [3]
[false] = [0]
[0]
[rm](x1, x2) = [0 1] x2 + [0]
[0 1] [0]
[nil] = [0]
[0]
[add](x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[1 0] [0 1] [0]
[ifrm](x1, x2, x3) = [1 0] x3 + [0]
[0 1] [0]
[eq^#](x1, x2) = [1 0] x2 + [0]
[0 0] [0]
[c_4](x1) = [1 0] x1 + [1]
[0 0] [0]
[rm^#](x1, x2) = [4 0] x1 + [0 7] x2 + [0]
[0 0] [0 0] [0]
[c_6](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[ifrm^#](x1, x2, x3) = [4 0] x2 + [2 5] x3 + [0]
[0 0] [0 0] [0]
[c_7](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_8](x1) = [1 1] x1 + [0]
[0 0] [0]
[purge^#](x1) = [3 4] x1 + [0]
[0 0] [0]
The order satisfies the following ordering constraints:
[eq(0(), 0())] = [0]
[0]
>= [0]
[0]
= [true()]
[eq(0(), s(X))] = [0]
[0]
>= [0]
[0]
= [false()]
[eq(s(X), 0())] = [0]
[0]
>= [0]
[0]
= [false()]
[eq(s(X), s(Y))] = [0]
[0]
>= [0]
[0]
= [eq(X, Y)]
[rm(N, nil())] = [0]
[0]
>= [0]
[0]
= [nil()]
[rm(N, add(M, X))] = [0 1] X + [1 0] M + [0]
[0 1] [1 0] [0]
>= [0 1] X + [0 0] M + [0]
[0 1] [1 0] [0]
= [ifrm(eq(N, M), N, add(M, X))]
[ifrm(true(), N, add(M, X))] = [0 1] X + [0 0] M + [0]
[0 1] [1 0] [0]
>= [0 1] X + [0]
[0 1] [0]
= [rm(N, X)]
[ifrm(false(), N, add(M, X))] = [0 1] X + [0 0] M + [0]
[0 1] [1 0] [0]
>= [0 1] X + [0 0] M + [0]
[0 1] [1 0] [0]
= [add(M, rm(N, X))]
[eq^#(s(X), s(Y))] = [1 0] Y + [4]
[0 0] [0]
> [1 0] Y + [1]
[0 0] [0]
= [c_4(eq^#(X, Y))]
[rm^#(N, add(M, X))] = [0 7] X + [4 0] N + [7 0] M + [0]
[0 0] [0 0] [0 0] [0]
>= [0 7] X + [4 0] N + [6 0] M + [0]
[0 0] [0 0] [0 0] [0]
= [c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))]
[ifrm^#(true(), N, add(M, X))] = [0 7] X + [4 0] N + [5 0] M + [0]
[0 0] [0 0] [0 0] [0]
>= [0 7] X + [4 0] N + [0]
[0 0] [0 0] [0]
= [c_7(rm^#(N, X))]
[ifrm^#(false(), N, add(M, X))] = [0 7] X + [4 0] N + [5 0] M + [0]
[0 0] [0 0] [0 0] [0]
>= [0 7] X + [4 0] N + [0]
[0 0] [0 0] [0]
= [c_8(rm^#(N, X))]
[purge^#(add(N, X))] = [0 7] X + [4 0] N + [0]
[0 0] [0 0] [0]
>= [0 7] X + [4 0] N + [0]
[0 0] [0 0] [0]
= [rm^#(N, X)]
[purge^#(add(N, X))] = [0 7] X + [4 0] N + [0]
[0 0] [0 0] [0]
>= [0 7] X + [0]
[0 0] [0]
= [purge^#(rm(N, X))]
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X)) }
Weak DPs:
{ eq^#(s(X), s(Y)) -> c_4(eq^#(X, Y))
, purge^#(add(N, X)) -> rm^#(N, X)
, purge^#(add(N, X)) -> purge^#(rm(N, X)) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ eq^#(s(X), s(Y)) -> c_4(eq^#(X, Y)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
, ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X)) }
Weak DPs:
{ purge^#(add(N, X)) -> rm^#(N, X)
, purge^#(add(N, X)) -> purge^#(rm(N, X)) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:
{ rm^#(N, add(M, X)) ->
c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ rm^#(N, add(M, X)) -> c_1(ifrm^#(eq(N, M), N, add(M, X)))
, ifrm^#(true(), N, add(M, X)) -> c_2(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_3(rm^#(N, X)) }
Weak DPs:
{ purge^#(add(N, X)) -> c_4(rm^#(N, X))
, purge^#(add(N, X)) -> c_5(purge^#(rm(N, X))) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 1: rm^#(N, add(M, X)) -> c_1(ifrm^#(eq(N, M), N, add(M, X)))
, 2: ifrm^#(true(), N, add(M, X)) -> c_2(rm^#(N, X))
, 3: ifrm^#(false(), N, add(M, X)) -> c_3(rm^#(N, X))
, 4: purge^#(add(N, X)) -> c_4(rm^#(N, X))
, 5: purge^#(add(N, X)) -> c_5(purge^#(rm(N, X))) }
Trs:
{ eq(0(), s(X)) -> false()
, ifrm(true(), N, add(M, X)) -> rm(N, X) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(c_5) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[eq](x1, x2) = [1] x1 + [0]
[0] = [1]
[true] = [1]
[s](x1) = [1] x1 + [0]
[false] = [0]
[rm](x1, x2) = [1] x2 + [0]
[nil] = [0]
[add](x1, x2) = [1] x2 + [2]
[ifrm](x1, x2, x3) = [1] x3 + [0]
[eq^#](x1, x2) = [0]
[c_4](x1) = [0]
[rm^#](x1, x2) = [6] x2 + [3]
[c_6](x1, x2) = [0]
[ifrm^#](x1, x2, x3) = [6] x3 + [0]
[c_7](x1) = [0]
[c_8](x1) = [0]
[purge^#](x1) = [6] x1 + [3]
[c] = [0]
[c_1](x1) = [1] x1 + [1]
[c_2](x1) = [1] x1 + [0]
[c_3](x1) = [1] x1 + [1]
[c_4](x1) = [1] x1 + [0]
[c_5](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[eq(0(), 0())] = [1]
>= [1]
= [true()]
[eq(0(), s(X))] = [1]
> [0]
= [false()]
[eq(s(X), 0())] = [1] X + [0]
>= [0]
= [false()]
[eq(s(X), s(Y))] = [1] X + [0]
>= [1] X + [0]
= [eq(X, Y)]
[rm(N, nil())] = [0]
>= [0]
= [nil()]
[rm(N, add(M, X))] = [1] X + [2]
>= [1] X + [2]
= [ifrm(eq(N, M), N, add(M, X))]
[ifrm(true(), N, add(M, X))] = [1] X + [2]
> [1] X + [0]
= [rm(N, X)]
[ifrm(false(), N, add(M, X))] = [1] X + [2]
>= [1] X + [2]
= [add(M, rm(N, X))]
[rm^#(N, add(M, X))] = [6] X + [15]
> [6] X + [13]
= [c_1(ifrm^#(eq(N, M), N, add(M, X)))]
[ifrm^#(true(), N, add(M, X))] = [6] X + [12]
> [6] X + [3]
= [c_2(rm^#(N, X))]
[ifrm^#(false(), N, add(M, X))] = [6] X + [12]
> [6] X + [4]
= [c_3(rm^#(N, X))]
[purge^#(add(N, X))] = [6] X + [15]
> [6] X + [3]
= [c_4(rm^#(N, X))]
[purge^#(add(N, X))] = [6] X + [15]
> [6] X + [3]
= [c_5(purge^#(rm(N, X)))]
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs:
{ rm^#(N, add(M, X)) -> c_1(ifrm^#(eq(N, M), N, add(M, X)))
, ifrm^#(true(), N, add(M, X)) -> c_2(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_3(rm^#(N, X))
, purge^#(add(N, X)) -> c_4(rm^#(N, X))
, purge^#(add(N, X)) -> c_5(purge^#(rm(N, X))) }
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ rm^#(N, add(M, X)) -> c_1(ifrm^#(eq(N, M), N, add(M, X)))
, ifrm^#(true(), N, add(M, X)) -> c_2(rm^#(N, X))
, ifrm^#(false(), N, add(M, X)) -> c_3(rm^#(N, X))
, purge^#(add(N, X)) -> c_4(rm^#(N, X))
, purge^#(add(N, X)) -> c_5(purge^#(rm(N, X))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ eq(0(), 0()) -> true()
, eq(0(), s(X)) -> false()
, eq(s(X), 0()) -> false()
, eq(s(X), s(Y)) -> eq(X, Y)
, rm(N, nil()) -> nil()
, rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
, ifrm(true(), N, add(M, X)) -> rm(N, X)
, ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
No rule is usable, rules are removed from the input problem.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^2))