We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ or(x, true()) -> true()
, or(true(), y) -> true()
, or(false(), false()) -> false()
, mem(x, nil()) -> false()
, mem(x, set(y)) -> =(x, y)
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
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(or) = {1, 2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[or](x1, x2) = [1] x1 + [1] x2 + [0]
[true] = [0]
[false] = [0]
[mem](x1, x2) = [1] x2 + [0]
[nil] = [0]
[set](x1) = [1] x1 + [5]
[=](x1, x2) = [0]
[union](x1, x2) = [1] x1 + [1] x2 + [0]
The order satisfies the following ordering constraints:
[or(x, true())] = [1] x + [0]
>= [0]
= [true()]
[or(true(), y)] = [1] y + [0]
>= [0]
= [true()]
[or(false(), false())] = [0]
>= [0]
= [false()]
[mem(x, nil())] = [0]
>= [0]
= [false()]
[mem(x, set(y))] = [1] y + [5]
> [0]
= [=(x, y)]
[mem(x, union(y, z))] = [1] y + [1] z + [0]
>= [1] y + [1] z + [0]
= [or(mem(x, y), mem(x, z))]
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:
{ or(x, true()) -> true()
, or(true(), y) -> true()
, or(false(), false()) -> false()
, mem(x, nil()) -> false()
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
Weak Trs: { mem(x, set(y)) -> =(x, y) }
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:
{ mem(x, nil()) -> false()
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
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(or) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[or](x1, x2) = [1] x1 + [1] x2 + [0]
[true] = [0]
[false] = [0]
[mem](x1, x2) = [4] x2 + [0]
[nil] = [2]
[set](x1) = [0]
[=](x1, x2) = [0]
[union](x1, x2) = [1] x1 + [1] x2 + [2]
The order satisfies the following ordering constraints:
[or(x, true())] = [1] x + [0]
>= [0]
= [true()]
[or(true(), y)] = [1] y + [0]
>= [0]
= [true()]
[or(false(), false())] = [0]
>= [0]
= [false()]
[mem(x, nil())] = [8]
> [0]
= [false()]
[mem(x, set(y))] = [0]
>= [0]
= [=(x, y)]
[mem(x, union(y, z))] = [4] y + [4] z + [8]
> [4] y + [4] z + [0]
= [or(mem(x, y), mem(x, z))]
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:
{ or(x, true()) -> true()
, or(true(), y) -> true()
, or(false(), false()) -> false() }
Weak Trs:
{ mem(x, nil()) -> false()
, mem(x, set(y)) -> =(x, y)
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
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(or) = {1, 2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[or](x1, x2) = [1] x1 + [1] x2 + [0]
[true] = [0]
[false] = [4]
[mem](x1, x2) = [1] x2 + [0]
[nil] = [4]
[set](x1) = [1] x1 + [0]
[=](x1, x2) = [0]
[union](x1, x2) = [1] x1 + [1] x2 + [1]
The order satisfies the following ordering constraints:
[or(x, true())] = [1] x + [0]
>= [0]
= [true()]
[or(true(), y)] = [1] y + [0]
>= [0]
= [true()]
[or(false(), false())] = [8]
> [4]
= [false()]
[mem(x, nil())] = [4]
>= [4]
= [false()]
[mem(x, set(y))] = [1] y + [0]
>= [0]
= [=(x, y)]
[mem(x, union(y, z))] = [1] y + [1] z + [1]
> [1] y + [1] z + [0]
= [or(mem(x, y), mem(x, z))]
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:
{ or(x, true()) -> true()
, or(true(), y) -> true() }
Weak Trs:
{ or(false(), false()) -> false()
, mem(x, nil()) -> false()
, mem(x, set(y)) -> =(x, y)
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
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(or) = {1, 2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[or](x1, x2) = [1] x1 + [1] x2 + [4]
[true] = [0]
[false] = [0]
[mem](x1, x2) = [1] x2 + [1]
[nil] = [4]
[set](x1) = [1] x1 + [0]
[=](x1, x2) = [0]
[union](x1, x2) = [1] x1 + [1] x2 + [7]
The order satisfies the following ordering constraints:
[or(x, true())] = [1] x + [4]
> [0]
= [true()]
[or(true(), y)] = [1] y + [4]
> [0]
= [true()]
[or(false(), false())] = [4]
> [0]
= [false()]
[mem(x, nil())] = [5]
> [0]
= [false()]
[mem(x, set(y))] = [1] y + [1]
> [0]
= [=(x, y)]
[mem(x, union(y, z))] = [1] y + [1] z + [8]
> [1] y + [1] z + [6]
= [or(mem(x, y), mem(x, z))]
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(1)).
Weak Trs:
{ or(x, true()) -> true()
, or(true(), y) -> true()
, or(false(), false()) -> false()
, mem(x, nil()) -> false()
, mem(x, set(y)) -> =(x, y)
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))