The Runtime Complexity (innermost) 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(true, x, y, z) → cond2(gr(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
add(0, x) → x
add(s(x), y) → s(add(x, y))
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr, add, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
add < cond2
cond2 = cond3
gr < cond3
add < cond3

(6) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
gr, cond1, cond2, add, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
add < cond2
cond2 = cond3
gr < cond3
add < cond3

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Induction Base:
gr(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
false

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) →_R^Ω(1)
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) →_IH
false

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
add, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
add < cond2
cond2 = cond3
add < cond3

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
add(gen_0':s4_0(n313_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n313_0, b)), rt ∈ Ω(1 + n313₀)

Induction Base:
add(gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
gen_0':s4_0(b)

Induction Step:
add(gen_0':s4_0(+(n313_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
s(add(gen_0':s4_0(n313_0), gen_0':s4_0(b))) →_IH
s(gen_0':s4_0(+(b, c314_0)))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
add(gen_0':s4_0(n313_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n313_0, b)), rt ∈ Ω(1 + n313₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
cond2, cond1, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol cond2.

(14) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
add(gen_0':s4_0(n313_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n313_0, b)), rt ∈ Ω(1 + n313₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
cond1, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
add(gen_0':s4_0(n313_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n313_0, b)), rt ∈ Ω(1 + n313₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol cond3.

(18) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
add(gen_0':s4_0(n313_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n313_0, b)), rt ∈ Ω(1 + n313₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
add(gen_0':s4_0(n313_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n313_0, b)), rt ∈ Ω(1 + n313₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) → cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) → cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) → cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) → cond1(gr(add(x, y), z), x, y, z)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
add(0', x) → x
add(s(x), y) → s(add(x, y))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
add :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

(26) BOUNDS(n^1, INF)