We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs:
{ a__f(X1, X2) -> f(X1, X2)
, a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
, mark(g(X)) -> g(mark(X))
, mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(g) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[a__f](x1, x2) = [1] x1 + [1]
[g](x1) = [1] x1 + [0]
[mark](x1) = [0]
[f](x1, x2) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[a__f(X1, X2)] = [1] X1 + [1]
> [1] X1 + [0]
= [f(X1, X2)]
[a__f(g(X), Y)] = [1] X + [1]
>= [1]
= [a__f(mark(X), f(g(X), Y))]
[mark(g(X))] = [0]
>= [0]
= [g(mark(X))]
[mark(f(X1, X2))] = [0]
? [1]
= [a__f(mark(X1), X2)]
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^2)).
Strict Trs:
{ a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
, mark(g(X)) -> g(mark(X))
, mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Weak Trs: { a__f(X1, X2) -> f(X1, X2) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(g) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[a__f](x1, x2) = [1] x1 + [4]
[g](x1) = [1] x1 + [4]
[mark](x1) = [1]
[f](x1, x2) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[a__f(X1, X2)] = [1] X1 + [4]
> [1] X1 + [0]
= [f(X1, X2)]
[a__f(g(X), Y)] = [1] X + [8]
> [5]
= [a__f(mark(X), f(g(X), Y))]
[mark(g(X))] = [1]
? [5]
= [g(mark(X))]
[mark(f(X1, X2))] = [1]
? [5]
= [a__f(mark(X1), X2)]
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^2)).
Strict Trs:
{ mark(g(X)) -> g(mark(X))
, mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Weak Trs:
{ a__f(X1, X2) -> f(X1, X2)
, a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
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^2)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(g) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[a__f](x1, x2) = [1 1] x1 + [4]
[0 1] [4]
[g](x1) = [1 1] x1 + [5]
[0 1] [4]
[mark](x1) = [1 1] x1 + [2]
[0 1] [3]
[f](x1, x2) = [1 1] x1 + [3]
[0 1] [4]
The order satisfies the following ordering constraints:
[a__f(X1, X2)] = [1 1] X1 + [4]
[0 1] [4]
> [1 1] X1 + [3]
[0 1] [4]
= [f(X1, X2)]
[a__f(g(X), Y)] = [1 2] X + [13]
[0 1] [8]
> [1 2] X + [9]
[0 1] [7]
= [a__f(mark(X), f(g(X), Y))]
[mark(g(X))] = [1 2] X + [11]
[0 1] [7]
> [1 2] X + [10]
[0 1] [7]
= [g(mark(X))]
[mark(f(X1, X2))] = [1 2] X1 + [9]
[0 1] [7]
>= [1 2] X1 + [9]
[0 1] [7]
= [a__f(mark(X1), X2)]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs: { mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Weak Trs:
{ a__f(X1, X2) -> f(X1, X2)
, a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
, mark(g(X)) -> g(mark(X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We use the processor 'matrix interpretation of dimension 2' to
orient following rules strictly.
Trs: { mark(f(X1, X2)) -> a__f(mark(X1), X2) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(g) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[a__f](x1, x2) = [1 4] x1 + [0]
[0 1] [1]
[g](x1) = [1 4] x1 + [0]
[0 1] [0]
[mark](x1) = [1 4] x1 + [0]
[0 1] [0]
[f](x1, x2) = [1 4] x1 + [0]
[0 1] [1]
The order satisfies the following ordering constraints:
[a__f(X1, X2)] = [1 4] X1 + [0]
[0 1] [1]
>= [1 4] X1 + [0]
[0 1] [1]
= [f(X1, X2)]
[a__f(g(X), Y)] = [1 8] X + [0]
[0 1] [1]
>= [1 8] X + [0]
[0 1] [1]
= [a__f(mark(X), f(g(X), Y))]
[mark(g(X))] = [1 8] X + [0]
[0 1] [0]
>= [1 8] X + [0]
[0 1] [0]
= [g(mark(X))]
[mark(f(X1, X2))] = [1 8] X1 + [4]
[0 1] [1]
> [1 8] X1 + [0]
[0 1] [1]
= [a__f(mark(X1), X2)]
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(X1, X2) -> f(X1, X2)
, a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
, mark(g(X)) -> g(mark(X))
, mark(f(X1, X2)) -> a__f(mark(X1), X2) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^2))