The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

eq(0, 0) → true
eq(0, s(Y)) → false
eq(s(X), 0) → false
eq(s(X), s(Y)) → eq(X, Y)
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0, nil)) → 0
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
eq(s(X), s(Y)) →⁺ eq(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:

eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
eq, le, min, replace, selsort

They will be analysed ascendingly in the following order:
eq < replace
eq < selsort
le < min
min < selsort
replace < selsort

(8) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
eq, le, min, replace, selsort

They will be analysed ascendingly in the following order:
eq < replace
eq < selsort
le < min
min < selsort
replace < selsort

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

Proved the following rewrite lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →_R^Ω(1)
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →_IH
true

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
le, min, replace, selsort

They will be analysed ascendingly in the following order:
le < min
min < selsort
replace < selsort

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) → true, rt ∈ Ω(1 + n576₀)

Induction Base:
le(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
le(gen_0':s4_0(+(n576_0, 1)), gen_0':s4_0(+(n576_0, 1))) →_R^Ω(1)
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) →_IH
true

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) → true, rt ∈ Ω(1 + n576₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
min, replace, selsort

They will be analysed ascendingly in the following order:
min < selsort
replace < selsort

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
min(gen_nil:cons5_0(+(1, n917_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n917₀)

Induction Base:
min(gen_nil:cons5_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
min(gen_nil:cons5_0(+(1, +(n917_0, 1)))) →_R^Ω(1)
ifmin(le(0', 0'), cons(0', cons(0', gen_nil:cons5_0(n917_0)))) →_L^Ω(1)
ifmin(true, cons(0', cons(0', gen_nil:cons5_0(n917_0)))) →_R^Ω(1)
min(cons(0', gen_nil:cons5_0(n917_0))) →_IH
gen_0':s4_0(0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) → true, rt ∈ Ω(1 + n576₀)
min(gen_nil:cons5_0(+(1, n917_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n917₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
replace, selsort

They will be analysed ascendingly in the following order:
replace < selsort

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) → true, rt ∈ Ω(1 + n576₀)
min(gen_nil:cons5_0(+(1, n917_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n917₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
selsort

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
selsort(gen_nil:cons5_0(n1470_0)) → gen_nil:cons5_0(n1470_0), rt ∈ Ω(1 + n1470₀ + n1470₀²)

Induction Base:
selsort(gen_nil:cons5_0(0)) →_R^Ω(1)
nil

Induction Step:
selsort(gen_nil:cons5_0(+(n1470_0, 1))) →_R^Ω(1)
ifselsort(eq(0', min(cons(0', gen_nil:cons5_0(n1470_0)))), cons(0', gen_nil:cons5_0(n1470_0))) →_L^{Ω(1 + n1470₀)}
ifselsort(eq(0', gen_0':s4_0(0)), cons(0', gen_nil:cons5_0(n1470_0))) →_L^Ω(1)
ifselsort(true, cons(0', gen_nil:cons5_0(n1470_0))) →_R^Ω(1)
cons(0', selsort(gen_nil:cons5_0(n1470_0))) →_IH
cons(0', gen_nil:cons5_0(c1471_0))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(21) Complex Obligation (BEST)

(22) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) → true, rt ∈ Ω(1 + n576₀)
min(gen_nil:cons5_0(+(1, n917_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n917₀)
selsort(gen_nil:cons5_0(n1470_0)) → gen_nil:cons5_0(n1470_0), rt ∈ Ω(1 + n1470₀ + n1470₀²)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
selsort(gen_nil:cons5_0(n1470_0)) → gen_nil:cons5_0(n1470_0), rt ∈ Ω(1 + n1470₀ + n1470₀²)

(24) BOUNDS(n^2, INF)

(25) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) → true, rt ∈ Ω(1 + n576₀)
min(gen_nil:cons5_0(+(1, n917_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n917₀)
selsort(gen_nil:cons5_0(n1470_0)) → gen_nil:cons5_0(n1470_0), rt ∈ Ω(1 + n1470₀ + n1470₀²)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
selsort(gen_nil:cons5_0(n1470_0)) → gen_nil:cons5_0(n1470_0), rt ∈ Ω(1 + n1470₀ + n1470₀²)

(27) BOUNDS(n^2, INF)

(28) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) → true, rt ∈ Ω(1 + n576₀)
min(gen_nil:cons5_0(+(1, n917_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n917₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(30) BOUNDS(n^1, INF)

(31) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
le(gen_0':s4_0(n576_0), gen_0':s4_0(n576_0)) → true, rt ∈ Ω(1 + n576₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(33) BOUNDS(n^1, INF)

(34) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(Y)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
le(0', Y) → true
le(s(X), 0') → false
le(s(X), s(Y)) → le(X, Y)
min(cons(0', nil)) → 0'
min(cons(s(N), nil)) → s(N)
min(cons(N, cons(M, L))) → ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) → min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) → min(cons(M, L))
replace(N, M, nil) → nil
replace(N, M, cons(K, L)) → ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) → cons(M, L)
ifrepl(false, N, M, cons(K, L)) → cons(K, replace(N, M, L))
selsort(nil) → nil
selsort(cons(N, L)) → ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) → cons(N, selsort(L))
ifselsort(false, cons(N, L)) → cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
le :: 0':s → 0':s → true:false
min :: nil:cons → 0':s
cons :: 0':s → nil:cons → nil:cons
nil :: nil:cons
ifmin :: true:false → nil:cons → 0':s
replace :: 0':s → 0':s → nil:cons → nil:cons
ifrepl :: true:false → 0':s → 0':s → nil:cons → nil:cons
selsort :: nil:cons → nil:cons
ifselsort :: true:false → nil:cons → nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(36) BOUNDS(n^1, INF)