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:

del(.(x, .(y, z))) → f(=(x, y), x, y, z)
f(true, x, y, z) → del(.(y, z))
f(false, x, y, z) → .(x, del(.(y, z)))
=(nil, nil) → true
=(.(x, y), nil) → false
=(nil, .(y, z)) → false
=(.(x, y), .(u, v)) → and(=(x, u), =(y, v))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

del(.(x, .(y, z))) → f(='(x, y), x, y, z)
f(true, x, y, z) → del(.(y, z))
f(false, x, y, z) → .(x, del(.(y, z)))
='(nil, nil) → true
='(.(x, y), nil) → false
='(nil, .(y, z)) → false
='(.(x, y), .(u, v)) → and(='(x, u), ='(y, v))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
del(.(x, .(y, z))) → f(='(x, y), x, y, z)
f(true, x, y, z) → del(.(y, z))
f(false, x, y, z) → .(x, del(.(y, z)))
='(nil, nil) → true
='(.(x, y), nil) → false
='(nil, .(y, z)) → false
='(.(x, y), .(u, v)) → and(='(x, u), ='(y, v))

Types:
del :: .:nil:u:v → .:nil:u:v
. :: .:nil:u:v → .:nil:u:v → .:nil:u:v
f :: true:false:and → .:nil:u:v → .:nil:u:v → .:nil:u:v → .:nil:u:v
=' :: .:nil:u:v → .:nil:u:v → true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and → true:false:and → true:false:and
hole_.:nil:u:v1_0 :: .:nil:u:v
hole_true:false:and2_0 :: true:false:and
gen_.:nil:u:v3_0 :: Nat → .:nil:u:v
gen_true:false:and4_0 :: Nat → true:false:and

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
del, ='

They will be analysed ascendingly in the following order:
=' < del

(8) Obligation:

TRS:
Rules:
del(.(x, .(y, z))) → f(='(x, y), x, y, z)
f(true, x, y, z) → del(.(y, z))
f(false, x, y, z) → .(x, del(.(y, z)))
='(nil, nil) → true
='(.(x, y), nil) → false
='(nil, .(y, z)) → false
='(.(x, y), .(u, v)) → and(='(x, u), ='(y, v))

Types:
del :: .:nil:u:v → .:nil:u:v
. :: .:nil:u:v → .:nil:u:v → .:nil:u:v
f :: true:false:and → .:nil:u:v → .:nil:u:v → .:nil:u:v → .:nil:u:v
=' :: .:nil:u:v → .:nil:u:v → true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and → true:false:and → true:false:and
hole_.:nil:u:v1_0 :: .:nil:u:v
hole_true:false:and2_0 :: true:false:and
gen_.:nil:u:v3_0 :: Nat → .:nil:u:v
gen_true:false:and4_0 :: Nat → true:false:and

Generator Equations:
gen_.:nil:u:v3_0(0) ⇔ nil
gen_.:nil:u:v3_0(+(x, 1)) ⇔ .(nil, gen_.:nil:u:v3_0(x))
gen_true:false:and4_0(0) ⇔ false
gen_true:false:and4_0(+(x, 1)) ⇔ and(false, gen_true:false:and4_0(x))

The following defined symbols remain to be analysed:
=', del

They will be analysed ascendingly in the following order:
=' < del

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol ='.

(10) Obligation:

TRS:
Rules:
del(.(x, .(y, z))) → f(='(x, y), x, y, z)
f(true, x, y, z) → del(.(y, z))
f(false, x, y, z) → .(x, del(.(y, z)))
='(nil, nil) → true
='(.(x, y), nil) → false
='(nil, .(y, z)) → false
='(.(x, y), .(u, v)) → and(='(x, u), ='(y, v))

Types:
del :: .:nil:u:v → .:nil:u:v
. :: .:nil:u:v → .:nil:u:v → .:nil:u:v
f :: true:false:and → .:nil:u:v → .:nil:u:v → .:nil:u:v → .:nil:u:v
=' :: .:nil:u:v → .:nil:u:v → true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and → true:false:and → true:false:and
hole_.:nil:u:v1_0 :: .:nil:u:v
hole_true:false:and2_0 :: true:false:and
gen_.:nil:u:v3_0 :: Nat → .:nil:u:v
gen_true:false:and4_0 :: Nat → true:false:and

Generator Equations:
gen_.:nil:u:v3_0(0) ⇔ nil
gen_.:nil:u:v3_0(+(x, 1)) ⇔ .(nil, gen_.:nil:u:v3_0(x))
gen_true:false:and4_0(0) ⇔ false
gen_true:false:and4_0(+(x, 1)) ⇔ and(false, gen_true:false:and4_0(x))

The following defined symbols remain to be analysed:
del

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

Proved the following rewrite lemma:
del(gen_.:nil:u:v3_0(+(2, n62_0))) → *5_0, rt ∈ Ω(n62₀)

Induction Base:
del(gen_.:nil:u:v3_0(+(2, 0)))

Induction Step:
del(gen_.:nil:u:v3_0(+(2, +(n62_0, 1)))) →_R^Ω(1)
f(='(nil, nil), nil, nil, gen_.:nil:u:v3_0(+(1, n62_0))) →_R^Ω(1)
f(true, nil, nil, gen_.:nil:u:v3_0(+(1, n62_0))) →_R^Ω(1)
del(.(nil, gen_.:nil:u:v3_0(+(1, n62_0)))) →_IH
*5_0

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
del(.(x, .(y, z))) → f(='(x, y), x, y, z)
f(true, x, y, z) → del(.(y, z))
f(false, x, y, z) → .(x, del(.(y, z)))
='(nil, nil) → true
='(.(x, y), nil) → false
='(nil, .(y, z)) → false
='(.(x, y), .(u, v)) → and(='(x, u), ='(y, v))

Types:
del :: .:nil:u:v → .:nil:u:v
. :: .:nil:u:v → .:nil:u:v → .:nil:u:v
f :: true:false:and → .:nil:u:v → .:nil:u:v → .:nil:u:v → .:nil:u:v
=' :: .:nil:u:v → .:nil:u:v → true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and → true:false:and → true:false:and
hole_.:nil:u:v1_0 :: .:nil:u:v
hole_true:false:and2_0 :: true:false:and
gen_.:nil:u:v3_0 :: Nat → .:nil:u:v
gen_true:false:and4_0 :: Nat → true:false:and

Lemmas:
del(gen_.:nil:u:v3_0(+(2, n62_0))) → *5_0, rt ∈ Ω(n62₀)

Generator Equations:
gen_.:nil:u:v3_0(0) ⇔ nil
gen_.:nil:u:v3_0(+(x, 1)) ⇔ .(nil, gen_.:nil:u:v3_0(x))
gen_true:false:and4_0(0) ⇔ false
gen_true:false:and4_0(+(x, 1)) ⇔ and(false, gen_true:false:and4_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
del(gen_.:nil:u:v3_0(+(2, n62_0))) → *5_0, rt ∈ Ω(n62₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
del(.(x, .(y, z))) → f(='(x, y), x, y, z)
f(true, x, y, z) → del(.(y, z))
f(false, x, y, z) → .(x, del(.(y, z)))
='(nil, nil) → true
='(.(x, y), nil) → false
='(nil, .(y, z)) → false
='(.(x, y), .(u, v)) → and(='(x, u), ='(y, v))

Types:
del :: .:nil:u:v → .:nil:u:v
. :: .:nil:u:v → .:nil:u:v → .:nil:u:v
f :: true:false:and → .:nil:u:v → .:nil:u:v → .:nil:u:v → .:nil:u:v
=' :: .:nil:u:v → .:nil:u:v → true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and → true:false:and → true:false:and
hole_.:nil:u:v1_0 :: .:nil:u:v
hole_true:false:and2_0 :: true:false:and
gen_.:nil:u:v3_0 :: Nat → .:nil:u:v
gen_true:false:and4_0 :: Nat → true:false:and

Lemmas:
del(gen_.:nil:u:v3_0(+(2, n62_0))) → *5_0, rt ∈ Ω(n62₀)

Generator Equations:
gen_.:nil:u:v3_0(0) ⇔ nil
gen_.:nil:u:v3_0(+(x, 1)) ⇔ .(nil, gen_.:nil:u:v3_0(x))
gen_true:false:and4_0(0) ⇔ false
gen_true:false:and4_0(+(x, 1)) ⇔ and(false, gen_true:false:and4_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
del(gen_.:nil:u:v3_0(+(2, n62_0))) → *5_0, rt ∈ Ω(n62₀)

(18) BOUNDS(n^1, INF)