* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
mem(x,nil()) -> false()
mem(x,set(y)) -> =(x,y)
mem(x,union(y,z)) -> or(mem(x,y),mem(x,z))
or(x,true()) -> true()
or(false(),false()) -> false()
or(true(),y) -> true()
- Signature:
{mem/2,or/2} / {=/2,false/0,nil/0,set/1,true/0,union/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {mem,or} and constructors {=,false,nil,set,true,union}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(or) = {1,2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(=) = [0]
p(false) = [3]
p(mem) = [4] x2 + [4]
p(nil) = [1]
p(or) = [1] x1 + [1] x2 + [12]
p(set) = [3]
p(true) = [8]
p(union) = [1] x1 + [1] x2 + [6]

Following rules are strictly oriented:
mem(x,nil()) = [8]
> [3]
= false()

mem(x,set(y)) = [16]
> [0]
= =(x,y)

mem(x,union(y,z)) = [4] y + [4] z + [28]
> [4] y + [4] z + [20]
= or(mem(x,y),mem(x,z))

or(x,true()) = [1] x + [20]
> [8]
= true()

or(false(),false()) = [18]
> [3]
= false()

or(true(),y) = [1] y + [20]
> [8]
= true()

Following rules are (at-least) weakly oriented:

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
mem(x,nil()) -> false()
mem(x,set(y)) -> =(x,y)
mem(x,union(y,z)) -> or(mem(x,y),mem(x,z))
or(x,true()) -> true()
or(false(),false()) -> false()
or(true(),y) -> true()
- Signature:
{mem/2,or/2} / {=/2,false/0,nil/0,set/1,true/0,union/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {mem,or} and constructors {=,false,nil,set,true,union}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))