YES(O(1),O(n^1))
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ a__f(X) -> f(X)
, a__f(f(a())) -> a__f(g(f(a())))
, mark(f(X)) -> a__f(X)
, mark(a()) -> a()
, mark(g(X)) -> g(mark(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.
Trs:
{ mark(f(X)) -> a__f(X)
, mark(a()) -> a() }
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:
----------
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[a__f](x1) = [2] x1 + [0]
[f](x1) = [1] x1 + [0]
[a] = [0]
[g](x1) = [1] x1 + [0]
[mark](x1) = [2] x1 + [1]
This order satisfies the following ordering constraints:
[a__f(X)] = [2] X + [0]
>= [1] X + [0]
= [f(X)]
[a__f(f(a()))] = [0]
>= [0]
= [a__f(g(f(a())))]
[mark(f(X))] = [2] X + [1]
> [2] X + [0]
= [a__f(X)]
[mark(a())] = [1]
> [0]
= [a()]
[mark(g(X))] = [2] X + [1]
>= [2] X + [1]
= [g(mark(X))]
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 Trs:
{ a__f(X) -> f(X)
, a__f(f(a())) -> a__f(g(f(a())))
, mark(g(X)) -> g(mark(X)) }
Weak Trs:
{ mark(f(X)) -> a__f(X)
, mark(a()) -> a() }
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: { a__f(X) -> f(X) }
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:
----------
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[a__f](x1) = [1] x1 + [1]
[f](x1) = [1] x1 + [0]
[a] = [0]
[g](x1) = [1] x1 + [0]
[mark](x1) = [1] x1 + [1]
This order satisfies the following ordering constraints:
[a__f(X)] = [1] X + [1]
> [1] X + [0]
= [f(X)]
[a__f(f(a()))] = [1]
>= [1]
= [a__f(g(f(a())))]
[mark(f(X))] = [1] X + [1]
>= [1] X + [1]
= [a__f(X)]
[mark(a())] = [1]
> [0]
= [a()]
[mark(g(X))] = [1] X + [1]
>= [1] X + [1]
= [g(mark(X))]
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 Trs:
{ a__f(f(a())) -> a__f(g(f(a())))
, mark(g(X)) -> g(mark(X)) }
Weak Trs:
{ a__f(X) -> f(X)
, mark(f(X)) -> a__f(X)
, mark(a()) -> a() }
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.
Trs: { a__f(f(a())) -> a__f(g(f(a()))) }
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:
----------
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).
[a__f](x1) = [1 1] x1 + [0]
[0 0] [1]
[f](x1) = [1 1] x1 + [0]
[0 0] [1]
[a] = [0]
[0]
[g](x1) = [1 0] x1 + [0]
[0 0] [0]
[mark](x1) = [1 0] x1 + [0]
[0 0] [1]
This order satisfies the following ordering constraints:
[a__f(X)] = [1 1] X + [0]
[0 0] [1]
>= [1 1] X + [0]
[0 0] [1]
= [f(X)]
[a__f(f(a()))] = [1]
[1]
> [0]
[1]
= [a__f(g(f(a())))]
[mark(f(X))] = [1 1] X + [0]
[0 0] [1]
>= [1 1] X + [0]
[0 0] [1]
= [a__f(X)]
[mark(a())] = [0]
[1]
>= [0]
[0]
= [a()]
[mark(g(X))] = [1 0] X + [0]
[0 0] [1]
>= [1 0] X + [0]
[0 0] [0]
= [g(mark(X))]
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 Trs: { mark(g(X)) -> g(mark(X)) }
Weak Trs:
{ a__f(X) -> f(X)
, a__f(f(a())) -> a__f(g(f(a())))
, mark(f(X)) -> a__f(X)
, mark(a()) -> a() }
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.
Trs: { mark(g(X)) -> g(mark(X)) }
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:
----------
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).
[a__f](x1) = [1 0] x1 + [0]
[0 0] [0]
[f](x1) = [1 0] x1 + [0]
[0 0] [0]
[a] = [0]
[0]
[g](x1) = [1 0] x1 + [0]
[0 1] [1]
[mark](x1) = [1 1] x1 + [0]
[0 1] [0]
This order satisfies the following ordering constraints:
[a__f(X)] = [1 0] X + [0]
[0 0] [0]
>= [1 0] X + [0]
[0 0] [0]
= [f(X)]
[a__f(f(a()))] = [0]
[0]
>= [0]
[0]
= [a__f(g(f(a())))]
[mark(f(X))] = [1 0] X + [0]
[0 0] [0]
>= [1 0] X + [0]
[0 0] [0]
= [a__f(X)]
[mark(a())] = [0]
[0]
>= [0]
[0]
= [a()]
[mark(g(X))] = [1 1] X + [1]
[0 1] [1]
> [1 1] X + [0]
[0 1] [1]
= [g(mark(X))]
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:
{ a__f(X) -> f(X)
, a__f(f(a())) -> a__f(g(f(a())))
, mark(f(X)) -> a__f(X)
, mark(a()) -> a()
, mark(g(X)) -> g(mark(X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))