The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

or(true, y) → true
or(x, true) → 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))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
mem(x, union(y, z)) →⁺ or(mem(x, y), mem(x, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [y / union(y, z)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

or(true, y) → true
or(x, true) → 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))

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
mem/0
set/0
='/0
='/1

(6) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(nil) → false
mem(set) → ='
mem(union(y, z)) → or(mem(y), mem(z))

S is empty.
Rewrite Strategy: FULL

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(8) Obligation:

TRS:
Rules:
or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(nil) → false
mem(set) → ='
mem(union(y, z)) → or(mem(y), mem(z))

Types:
or :: true:false:=' → true:false:=' → true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union → true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union → nil:set:union → nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat → nil:set:union

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mem

(10) Obligation:

TRS:
Rules:
or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(nil) → false
mem(set) → ='
mem(union(y, z)) → or(mem(y), mem(z))

Types:
or :: true:false:=' → true:false:=' → true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union → true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union → nil:set:union → nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat → nil:set:union

Generator Equations:
gen_nil:set:union3_0(0) ⇔ nil
gen_nil:set:union3_0(+(x, 1)) ⇔ union(nil, gen_nil:set:union3_0(x))

The following defined symbols remain to be analysed:
mem

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mem(gen_nil:set:union3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Induction Base:
mem(gen_nil:set:union3_0(0)) →_R^Ω(1)
false

Induction Step:
mem(gen_nil:set:union3_0(+(n5_0, 1))) →_R^Ω(1)
or(mem(nil), mem(gen_nil:set:union3_0(n5_0))) →_R^Ω(1)
or(false, mem(gen_nil:set:union3_0(n5_0))) →_IH
or(false, false) →_R^Ω(1)
false

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(nil) → false
mem(set) → ='
mem(union(y, z)) → or(mem(y), mem(z))

Types:
or :: true:false:=' → true:false:=' → true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union → true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union → nil:set:union → nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat → nil:set:union

Lemmas:
mem(gen_nil:set:union3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:set:union3_0(0) ⇔ nil
gen_nil:set:union3_0(+(x, 1)) ⇔ union(nil, gen_nil:set:union3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mem(gen_nil:set:union3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(nil) → false
mem(set) → ='
mem(union(y, z)) → or(mem(y), mem(z))

Types:
or :: true:false:=' → true:false:=' → true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union → true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union → nil:set:union → nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat → nil:set:union

Lemmas:
mem(gen_nil:set:union3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:set:union3_0(0) ⇔ nil
gen_nil:set:union3_0(+(x, 1)) ⇔ union(nil, gen_nil:set:union3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mem(gen_nil:set:union3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)