The Runtime Complexity (innermost) 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:

purge(nil) → nil
purge(.(x, y)) → .(x, purge(remove(x, y)))
remove(x, nil) → nil
remove(x, .(y, z)) → if(=(x, y), remove(x, z), .(y, remove(x, z)))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(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:

purge(nil) → nil
purge(.(x, y)) → .(x, purge(remove(x, y)))
remove(x, nil) → nil
remove(x, .(y, z)) → if(='(x, y), remove(x, z), .(y, remove(x, z)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
purge(nil) → nil
purge(.(x, y)) → .(x, purge(remove(x, y)))
remove(x, nil) → nil
remove(x, .(y, z)) → if(='(x, y), remove(x, z), .(y, remove(x, z)))

Types:
purge :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: a → nil:.:if → nil:.:if
remove :: a → nil:.:if → nil:.:if
if :: =' → nil:.:if → nil:.:if → nil:.:if
=' :: a → a → ='
hole_nil:.:if1_0 :: nil:.:if
hole_a2_0 :: a
hole_='3_0 :: ='
gen_nil:.:if4_0 :: Nat → nil:.:if

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

Heuristically decided to analyse the following defined symbols:
purge, remove

They will be analysed ascendingly in the following order:
remove < purge

(8) Obligation:

Innermost TRS:
Rules:
purge(nil) → nil
purge(.(x, y)) → .(x, purge(remove(x, y)))
remove(x, nil) → nil
remove(x, .(y, z)) → if(='(x, y), remove(x, z), .(y, remove(x, z)))

Types:
purge :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: a → nil:.:if → nil:.:if
remove :: a → nil:.:if → nil:.:if
if :: =' → nil:.:if → nil:.:if → nil:.:if
=' :: a → a → ='
hole_nil:.:if1_0 :: nil:.:if
hole_a2_0 :: a
hole_='3_0 :: ='
gen_nil:.:if4_0 :: Nat → nil:.:if

Generator Equations:
gen_nil:.:if4_0(0) ⇔ nil
gen_nil:.:if4_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.:if4_0(x))

The following defined symbols remain to be analysed:
remove, purge

They will be analysed ascendingly in the following order:
remove < purge

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

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

(10) Obligation:

Innermost TRS:
Rules:
purge(nil) → nil
purge(.(x, y)) → .(x, purge(remove(x, y)))
remove(x, nil) → nil
remove(x, .(y, z)) → if(='(x, y), remove(x, z), .(y, remove(x, z)))

Types:
purge :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: a → nil:.:if → nil:.:if
remove :: a → nil:.:if → nil:.:if
if :: =' → nil:.:if → nil:.:if → nil:.:if
=' :: a → a → ='
hole_nil:.:if1_0 :: nil:.:if
hole_a2_0 :: a
hole_='3_0 :: ='
gen_nil:.:if4_0 :: Nat → nil:.:if

Generator Equations:
gen_nil:.:if4_0(0) ⇔ nil
gen_nil:.:if4_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.:if4_0(x))

The following defined symbols remain to be analysed:
purge

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

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

(12) Obligation:

Innermost TRS:
Rules:
purge(nil) → nil
purge(.(x, y)) → .(x, purge(remove(x, y)))
remove(x, nil) → nil
remove(x, .(y, z)) → if(='(x, y), remove(x, z), .(y, remove(x, z)))

Types:
purge :: nil:.:if → nil:.:if
nil :: nil:.:if
. :: a → nil:.:if → nil:.:if
remove :: a → nil:.:if → nil:.:if
if :: =' → nil:.:if → nil:.:if → nil:.:if
=' :: a → a → ='
hole_nil:.:if1_0 :: nil:.:if
hole_a2_0 :: a
hole_='3_0 :: ='
gen_nil:.:if4_0 :: Nat → nil:.:if

Generator Equations:
gen_nil:.:if4_0(0) ⇔ nil
gen_nil:.:if4_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.:if4_0(x))

No more defined symbols left to analyse.