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:

f(true, x, y, z) → g(gt(x, y), x, y, z)
g(true, x, y, z) → f(gt(x, z), x, s(y), z)
g(true, x, y, z) → f(gt(x, z), x, y, s(z))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

f(true, x, y, z) → g(gt(x, y), x, y, z)
g(true, x, y, z) → f(gt(x, z), x, s(y), z)
g(true, x, y, z) → f(gt(x, z), x, y, s(z))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(true, x, y, z) → g(gt(x, y), x, y, z)
g(true, x, y, z) → f(gt(x, z), x, s(y), z)
g(true, x, y, z) → f(gt(x, z), x, y, s(z))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
f :: true:false → s:0' → s:0' → s:0' → f:g
true :: true:false
g :: true:false → s:0' → s:0' → s:0' → f:g
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_f:g1_0 :: f:g
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
f, g, gt

They will be analysed ascendingly in the following order:
f = g
gt < f
gt < g

(8) Obligation:

TRS:
Rules:
f(true, x, y, z) → g(gt(x, y), x, y, z)
g(true, x, y, z) → f(gt(x, z), x, s(y), z)
g(true, x, y, z) → f(gt(x, z), x, y, s(z))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
f :: true:false → s:0' → s:0' → s:0' → f:g
true :: true:false
g :: true:false → s:0' → s:0' → s:0' → f:g
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_f:g1_0 :: f:g
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
gt, f, g

They will be analysed ascendingly in the following order:
f = g
gt < f
gt < g

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

Proved the following rewrite lemma:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Induction Base:
gt(gen_s:0'4_0(0), gen_s:0'4_0(0)) →_R^Ω(1)
false

Induction Step:
gt(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) →_R^Ω(1)
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) →_IH
false

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
f(true, x, y, z) → g(gt(x, y), x, y, z)
g(true, x, y, z) → f(gt(x, z), x, s(y), z)
g(true, x, y, z) → f(gt(x, z), x, y, s(z))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
f :: true:false → s:0' → s:0' → s:0' → f:g
true :: true:false
g :: true:false → s:0' → s:0' → s:0' → f:g
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_f:g1_0 :: f:g
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
g, f

They will be analysed ascendingly in the following order:
f = g

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol g.

(13) Obligation:

TRS:
Rules:
f(true, x, y, z) → g(gt(x, y), x, y, z)
g(true, x, y, z) → f(gt(x, z), x, s(y), z)
g(true, x, y, z) → f(gt(x, z), x, y, s(z))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
f :: true:false → s:0' → s:0' → s:0' → f:g
true :: true:false
g :: true:false → s:0' → s:0' → s:0' → f:g
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_f:g1_0 :: f:g
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
f

They will be analysed ascendingly in the following order:
f = g

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(15) Obligation:

TRS:
Rules:
f(true, x, y, z) → g(gt(x, y), x, y, z)
g(true, x, y, z) → f(gt(x, z), x, s(y), z)
g(true, x, y, z) → f(gt(x, z), x, y, s(z))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
f :: true:false → s:0' → s:0' → s:0' → f:g
true :: true:false
g :: true:false → s:0' → s:0' → s:0' → f:g
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_f:g1_0 :: f:g
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
f(true, x, y, z) → g(gt(x, y), x, y, z)
g(true, x, y, z) → f(gt(x, z), x, s(y), z)
g(true, x, y, z) → f(gt(x, z), x, y, s(z))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)

Types:
f :: true:false → s:0' → s:0' → s:0' → f:g
true :: true:false
g :: true:false → s:0' → s:0' → s:0' → f:g
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
false :: true:false
hole_f:g1_0 :: f:g
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

(20) BOUNDS(n^1, INF)