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:

cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
p(0) → 0
p(s(x)) → x

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
p(0') → 0'
p(s(x)) → x

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: s:0' → s:0' → cond1:cond2
s :: s:0' → s:0'
cond2 :: true:false → s:0' → s:0' → cond1:cond2
gr :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
p :: s:0' → s:0'
0' :: s:0'
neq :: s:0' → s:0' → true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
cond1, gr, neq

They will be analysed ascendingly in the following order:
gr < cond1

(8) Obligation:

TRS:
Rules:
cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: s:0' → s:0' → cond1:cond2
s :: s:0' → s:0'
cond2 :: true:false → s:0' → s:0' → cond1:cond2
gr :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
p :: s:0' → s:0'
0' :: s:0'
neq :: s:0' → s:0' → true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
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:
gr, cond1, neq

They will be analysed ascendingly in the following order:
gr < cond1

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

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

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

Induction Step:
gr(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) →_R^Ω(1)
gr(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:
cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: s:0' → s:0' → cond1:cond2
s :: s:0' → s:0'
cond2 :: true:false → s:0' → s:0' → cond1:cond2
gr :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
p :: s:0' → s:0'
0' :: s:0'
neq :: s:0' → s:0' → true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gr(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:
cond1, neq

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

Could not prove a rewrite lemma for the defined symbol cond1.

(13) Obligation:

TRS:
Rules:
cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: s:0' → s:0' → cond1:cond2
s :: s:0' → s:0'
cond2 :: true:false → s:0' → s:0' → cond1:cond2
gr :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
p :: s:0' → s:0'
0' :: s:0'
neq :: s:0' → s:0' → true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gr(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:
neq

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
neq(gen_s:0'4_0(n15112_0), gen_s:0'4_0(n15112_0)) → false, rt ∈ Ω(1 + n15112₀)

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

Induction Step:
neq(gen_s:0'4_0(+(n15112_0, 1)), gen_s:0'4_0(+(n15112_0, 1))) →_R^Ω(1)
neq(gen_s:0'4_0(n15112_0), gen_s:0'4_0(n15112_0)) →_IH
false

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

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: s:0' → s:0' → cond1:cond2
s :: s:0' → s:0'
cond2 :: true:false → s:0' → s:0' → cond1:cond2
gr :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
p :: s:0' → s:0'
0' :: s:0'
neq :: s:0' → s:0' → true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
neq(gen_s:0'4_0(n15112_0), gen_s:0'4_0(n15112_0)) → false, rt ∈ Ω(1 + n15112₀)

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.

(17) LowerBoundsProof (EQUIVALENT transformation)

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

(18) BOUNDS(n^1, INF)

(19) Obligation:

TRS:
Rules:
cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: s:0' → s:0' → cond1:cond2
s :: s:0' → s:0'
cond2 :: true:false → s:0' → s:0' → cond1:cond2
gr :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
p :: s:0' → s:0'
0' :: s:0'
neq :: s:0' → s:0' → true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
neq(gen_s:0'4_0(n15112_0), gen_s:0'4_0(n15112_0)) → false, rt ∈ Ω(1 + n15112₀)

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.

(20) LowerBoundsProof (EQUIVALENT transformation)

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

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
cond1(s(x), y) → cond2(gr(s(x), y), s(x), y)
cond2(true, x, y) → cond1(y, y)
cond2(false, x, y) → cond1(p(x), y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: s:0' → s:0' → cond1:cond2
s :: s:0' → s:0'
cond2 :: true:false → s:0' → s:0' → cond1:cond2
gr :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
p :: s:0' → s:0'
0' :: s:0'
neq :: s:0' → s:0' → true:false
hole_cond1:cond21_0 :: cond1:cond2
hole_s:0'2_0 :: s:0'
hole_true:false3_0 :: true:false
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
gr(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.

(23) LowerBoundsProof (EQUIVALENT transformation)

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

(24) BOUNDS(n^1, INF)