We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, activate(X) -> X
, plus(N, s(M)) -> U11(tt(), M, N)
, plus(N, 0()) -> N }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The input is overlay and right-linear. Switching to innermost
rewriting.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, activate(X) -> X
, plus(N, s(M)) -> U11(tt(), M, N)
, plus(N, 0()) -> N }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U12) = {2, 3}, Uargs(s) = {1}, Uargs(plus) = {1, 2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3) = [1] x2 + [1] x3 + [1]
[tt] = [0]
[U12](x1, x2, x3) = [1] x2 + [1] x3 + [0]
[activate](x1) = [1] x1 + [1]
[s](x1) = [1] x1 + [0]
[plus](x1, x2) = [1] x1 + [1] x2 + [0]
[0] = [5]
The order satisfies the following ordering constraints:
[U11(tt(), M, N)] = [1] M + [1] N + [1]
? [1] M + [1] N + [2]
= [U12(tt(), activate(M), activate(N))]
[U12(tt(), M, N)] = [1] M + [1] N + [0]
? [1] M + [1] N + [2]
= [s(plus(activate(N), activate(M)))]
[activate(X)] = [1] X + [1]
> [1] X + [0]
= [X]
[plus(N, s(M))] = [1] M + [1] N + [0]
? [1] M + [1] N + [1]
= [U11(tt(), M, N)]
[plus(N, 0())] = [1] N + [5]
> [1] N + [0]
= [N]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, s(M)) -> U11(tt(), M, N) }
Weak Trs:
{ activate(X) -> X
, plus(N, 0()) -> N }
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.
Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, s(M)) -> U11(tt(), M, N) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are usable:
Uargs(U12) = {2, 3}, Uargs(s) = {1}, Uargs(plus) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[U11](x1, x2, x3) = [4] x1 + [2] x2 + [3] x3 + [6]
[tt] = [1]
[U12](x1, x2, x3) = [5] x1 + [2] x2 + [3] x3 + [3]
[activate](x1) = [1] x1 + [0]
[s](x1) = [1] x1 + [5]
[plus](x1, x2) = [3] x1 + [2] x2 + [2]
[0] = [0]
The order satisfies the following ordering constraints:
[U11(tt(), M, N)] = [2] M + [3] N + [10]
> [2] M + [3] N + [8]
= [U12(tt(), activate(M), activate(N))]
[U12(tt(), M, N)] = [2] M + [3] N + [8]
> [2] M + [3] N + [7]
= [s(plus(activate(N), activate(M)))]
[activate(X)] = [1] X + [0]
>= [1] X + [0]
= [X]
[plus(N, s(M))] = [2] M + [3] N + [12]
> [2] M + [3] N + [10]
= [U11(tt(), M, N)]
[plus(N, 0())] = [3] N + [2]
> [1] N + [0]
= [N]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, activate(X) -> X
, plus(N, s(M)) -> U11(tt(), M, N)
, plus(N, 0()) -> N }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))