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:

implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(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
implies(not(x), or(not(x19_0), z)) →⁺ implies(x, or(x19_0, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / not(x), x19_0 / not(x19_0)].
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:

implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))

Types:
implies :: not → or → or
not :: not → not
or :: not → or → or
hole_or1_0 :: or
hole_not2_0 :: not
gen_or3_0 :: Nat → or
gen_not4_0 :: Nat → not

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

Heuristically decided to analyse the following defined symbols:
implies

(8) Obligation:

TRS:
Rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))

Types:
implies :: not → or → or
not :: not → not
or :: not → or → or
hole_or1_0 :: or
hole_not2_0 :: not
gen_or3_0 :: Nat → or
gen_not4_0 :: Nat → not

Generator Equations:
gen_or3_0(0) ⇔ hole_or1_0
gen_or3_0(+(x, 1)) ⇔ or(hole_not2_0, gen_or3_0(x))
gen_not4_0(0) ⇔ hole_not2_0
gen_not4_0(+(x, 1)) ⇔ not(gen_not4_0(x))

The following defined symbols remain to be analysed:
implies

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
implies(gen_not4_0(1), gen_or3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

Induction Base:
implies(gen_not4_0(1), gen_or3_0(0))

Induction Step:
implies(gen_not4_0(1), gen_or3_0(+(n6_0, 1))) →_R^Ω(1)
or(hole_not2_0, implies(gen_not4_0(1), gen_or3_0(n6_0))) →_IH
or(hole_not2_0, *5_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))

Types:
implies :: not → or → or
not :: not → not
or :: not → or → or
hole_or1_0 :: or
hole_not2_0 :: not
gen_or3_0 :: Nat → or
gen_not4_0 :: Nat → not

Lemmas:
implies(gen_not4_0(1), gen_or3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

Generator Equations:
gen_or3_0(0) ⇔ hole_or1_0
gen_or3_0(+(x, 1)) ⇔ or(hole_not2_0, gen_or3_0(x))
gen_not4_0(0) ⇔ hole_not2_0
gen_not4_0(+(x, 1)) ⇔ not(gen_not4_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
implies(gen_not4_0(1), gen_or3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))

Types:
implies :: not → or → or
not :: not → not
or :: not → or → or
hole_or1_0 :: or
hole_not2_0 :: not
gen_or3_0 :: Nat → or
gen_not4_0 :: Nat → not

Lemmas:
implies(gen_not4_0(1), gen_or3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

Generator Equations:
gen_or3_0(0) ⇔ hole_or1_0
gen_or3_0(+(x, 1)) ⇔ or(hole_not2_0, gen_or3_0(x))
gen_not4_0(0) ⇔ hole_not2_0
gen_not4_0(+(x, 1)) ⇔ not(gen_not4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
implies(gen_not4_0(1), gen_or3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

(16) BOUNDS(n^1, INF)