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

(0) Obligation:

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

msort(nil) → nil
msort(.(x, y)) → .(min(x, y), msort(del(min(x, y), .(x, y))))
min(x, nil) → x
min(x, .(y, z)) → if(<=(x, y), min(x, z), min(y, z))
del(x, nil) → nil
del(x, .(y, z)) → if(=(x, y), z, .(y, del(x, z)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
min(x, .(y, z)) →⁺ if(<=(x, y), min(x, z), min(y, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [z / .(y, z)].
The result substitution is [ ].

The rewrite sequence
min(x, .(y, z)) →⁺ if(<=(x, y), min(x, z), min(y, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
The pumping substitution is [z / .(y, z)].
The result substitution is [x / y].

(2) BOUNDS(2^n, 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:

msort(nil) → nil
msort(.(x, y)) → .(min(x, y), msort(del(min(x, y), .(x, y))))
min(x, nil) → x
min(x, .(y, z)) → if(<=(x, y), min(x, z), min(y, z))
del(x, nil) → nil
del(x, .(y, z)) → if(='(x, y), z, .(y, del(x, z)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
msort(nil) → nil
msort(.(x, y)) → .(min(x, y), msort(del(min(x, y), .(x, y))))
min(x, nil) → x
min(x, .(y, z)) → if(<=(x, y), min(x, z), min(y, z))
del(x, nil) → nil
del(x, .(y, z)) → if(='(x, y), z, .(y, del(x, z)))

Types:
msort :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: nil:.:if → nil:.:if → nil:.:if
min :: nil:.:if → nil:.:if → nil:.:if
del :: nil:.:if → nil:.:if → nil:.:if
if :: <=:=' → nil:.:if → nil:.:if → nil:.:if
<= :: nil:.:if → nil:.:if → <=:='
=' :: nil:.:if → nil:.:if → <=:='
hole_nil:.:if1_0 :: nil:.:if
hole_<=:='2_0 :: <=:='
gen_nil:.:if3_0 :: Nat → nil:.:if

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

Heuristically decided to analyse the following defined symbols:
msort, min, del

They will be analysed ascendingly in the following order:
min < msort
del < msort

(8) Obligation:

TRS:
Rules:
msort(nil) → nil
msort(.(x, y)) → .(min(x, y), msort(del(min(x, y), .(x, y))))
min(x, nil) → x
min(x, .(y, z)) → if(<=(x, y), min(x, z), min(y, z))
del(x, nil) → nil
del(x, .(y, z)) → if(='(x, y), z, .(y, del(x, z)))

Types:
msort :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: nil:.:if → nil:.:if → nil:.:if
min :: nil:.:if → nil:.:if → nil:.:if
del :: nil:.:if → nil:.:if → nil:.:if
if :: <=:=' → nil:.:if → nil:.:if → nil:.:if
<= :: nil:.:if → nil:.:if → <=:='
=' :: nil:.:if → nil:.:if → <=:='
hole_nil:.:if1_0 :: nil:.:if
hole_<=:='2_0 :: <=:='
gen_nil:.:if3_0 :: Nat → nil:.:if

Generator Equations:
gen_nil:.:if3_0(0) ⇔ nil
gen_nil:.:if3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if3_0(x))

The following defined symbols remain to be analysed:
min, msort, del

They will be analysed ascendingly in the following order:
min < msort
del < msort

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
msort(nil) → nil
msort(.(x, y)) → .(min(x, y), msort(del(min(x, y), .(x, y))))
min(x, nil) → x
min(x, .(y, z)) → if(<=(x, y), min(x, z), min(y, z))
del(x, nil) → nil
del(x, .(y, z)) → if(='(x, y), z, .(y, del(x, z)))

Types:
msort :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: nil:.:if → nil:.:if → nil:.:if
min :: nil:.:if → nil:.:if → nil:.:if
del :: nil:.:if → nil:.:if → nil:.:if
if :: <=:=' → nil:.:if → nil:.:if → nil:.:if
<= :: nil:.:if → nil:.:if → <=:='
=' :: nil:.:if → nil:.:if → <=:='
hole_nil:.:if1_0 :: nil:.:if
hole_<=:='2_0 :: <=:='
gen_nil:.:if3_0 :: Nat → nil:.:if

Generator Equations:
gen_nil:.:if3_0(0) ⇔ nil
gen_nil:.:if3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if3_0(x))

The following defined symbols remain to be analysed:
del, msort

They will be analysed ascendingly in the following order:
del < msort

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
msort(nil) → nil
msort(.(x, y)) → .(min(x, y), msort(del(min(x, y), .(x, y))))
min(x, nil) → x
min(x, .(y, z)) → if(<=(x, y), min(x, z), min(y, z))
del(x, nil) → nil
del(x, .(y, z)) → if(='(x, y), z, .(y, del(x, z)))

Types:
msort :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: nil:.:if → nil:.:if → nil:.:if
min :: nil:.:if → nil:.:if → nil:.:if
del :: nil:.:if → nil:.:if → nil:.:if
if :: <=:=' → nil:.:if → nil:.:if → nil:.:if
<= :: nil:.:if → nil:.:if → <=:='
=' :: nil:.:if → nil:.:if → <=:='
hole_nil:.:if1_0 :: nil:.:if
hole_<=:='2_0 :: <=:='
gen_nil:.:if3_0 :: Nat → nil:.:if

Generator Equations:
gen_nil:.:if3_0(0) ⇔ nil
gen_nil:.:if3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if3_0(x))

The following defined symbols remain to be analysed:
msort

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

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

(14) Obligation:

TRS:
Rules:
msort(nil) → nil
msort(.(x, y)) → .(min(x, y), msort(del(min(x, y), .(x, y))))
min(x, nil) → x
min(x, .(y, z)) → if(<=(x, y), min(x, z), min(y, z))
del(x, nil) → nil
del(x, .(y, z)) → if(='(x, y), z, .(y, del(x, z)))

Types:
msort :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: nil:.:if → nil:.:if → nil:.:if
min :: nil:.:if → nil:.:if → nil:.:if
del :: nil:.:if → nil:.:if → nil:.:if
if :: <=:=' → nil:.:if → nil:.:if → nil:.:if
<= :: nil:.:if → nil:.:if → <=:='
=' :: nil:.:if → nil:.:if → <=:='
hole_nil:.:if1_0 :: nil:.:if
hole_<=:='2_0 :: <=:='
gen_nil:.:if3_0 :: Nat → nil:.:if

Generator Equations:
gen_nil:.:if3_0(0) ⇔ nil
gen_nil:.:if3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if3_0(x))

No more defined symbols left to analyse.