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:

circ(cons(a, s), t) → cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) → cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) → cons(lift, circ(s, t))
circ(circ(s, t), u) → circ(s, circ(t, u))
circ(s, id) → s
circ(id, s) → s
circ(cons(lift, s), circ(cons(lift, t), u)) → circ(cons(lift, circ(s, t)), u)
subst(a, id) → a
msubst(a, id) → a
msubst(msubst(a, s), t) → msubst(a, circ(s, t))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
circ(cons(a, s), t) →⁺ cons(msubst(a, t), circ(s, t))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [s / cons(a, s)].
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:

circ(cons(a, s), t) → cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) → cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) → cons(lift, circ(s, t))
circ(circ(s, t), u) → circ(s, circ(t, u))
circ(s, id) → s
circ(id, s) → s
circ(cons(lift, s), circ(cons(lift, t), u)) → circ(cons(lift, circ(s, t)), u)
subst(a, id) → a
msubst(a, id) → a
msubst(msubst(a, s), t) → msubst(a, circ(s, t))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
circ(cons(a, s), t) → cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) → cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) → cons(lift, circ(s, t))
circ(circ(s, t), u) → circ(s, circ(t, u))
circ(s, id) → s
circ(id, s) → s
circ(cons(lift, s), circ(cons(lift, t), u)) → circ(cons(lift, circ(s, t)), u)
subst(a, id) → a
msubst(a, id) → a
msubst(msubst(a, s), t) → msubst(a, circ(s, t))

Types:
circ :: cons:id → cons:id → cons:id
cons :: lift → cons:id → cons:id
msubst :: lift → cons:id → lift
lift :: lift
id :: cons:id
subst :: subst → cons:id → subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat → cons:id

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

Heuristically decided to analyse the following defined symbols:
circ, msubst

They will be analysed ascendingly in the following order:
circ = msubst

(8) Obligation:

TRS:
Rules:
circ(cons(a, s), t) → cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) → cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) → cons(lift, circ(s, t))
circ(circ(s, t), u) → circ(s, circ(t, u))
circ(s, id) → s
circ(id, s) → s
circ(cons(lift, s), circ(cons(lift, t), u)) → circ(cons(lift, circ(s, t)), u)
subst(a, id) → a
msubst(a, id) → a
msubst(msubst(a, s), t) → msubst(a, circ(s, t))

Types:
circ :: cons:id → cons:id → cons:id
cons :: lift → cons:id → cons:id
msubst :: lift → cons:id → lift
lift :: lift
id :: cons:id
subst :: subst → cons:id → subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat → cons:id

Generator Equations:
gen_cons:id4_0(0) ⇔ id
gen_cons:id4_0(+(x, 1)) ⇔ cons(lift, gen_cons:id4_0(x))

The following defined symbols remain to be analysed:
msubst, circ

They will be analysed ascendingly in the following order:
circ = msubst

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

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

(10) Obligation:

TRS:
Rules:
circ(cons(a, s), t) → cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) → cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) → cons(lift, circ(s, t))
circ(circ(s, t), u) → circ(s, circ(t, u))
circ(s, id) → s
circ(id, s) → s
circ(cons(lift, s), circ(cons(lift, t), u)) → circ(cons(lift, circ(s, t)), u)
subst(a, id) → a
msubst(a, id) → a
msubst(msubst(a, s), t) → msubst(a, circ(s, t))

Types:
circ :: cons:id → cons:id → cons:id
cons :: lift → cons:id → cons:id
msubst :: lift → cons:id → lift
lift :: lift
id :: cons:id
subst :: subst → cons:id → subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat → cons:id

Generator Equations:
gen_cons:id4_0(0) ⇔ id
gen_cons:id4_0(+(x, 1)) ⇔ cons(lift, gen_cons:id4_0(x))

The following defined symbols remain to be analysed:
circ

They will be analysed ascendingly in the following order:
circ = msubst

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0)) → gen_cons:id4_0(n16_0), rt ∈ Ω(1 + n16₀)

Induction Base:
circ(gen_cons:id4_0(0), gen_cons:id4_0(0)) →_R^Ω(1)
gen_cons:id4_0(0)

Induction Step:
circ(gen_cons:id4_0(+(n16_0, 1)), gen_cons:id4_0(+(n16_0, 1))) →_R^Ω(1)
cons(lift, circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0))) →_IH
cons(lift, gen_cons:id4_0(c17_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
circ(cons(a, s), t) → cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) → cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) → cons(lift, circ(s, t))
circ(circ(s, t), u) → circ(s, circ(t, u))
circ(s, id) → s
circ(id, s) → s
circ(cons(lift, s), circ(cons(lift, t), u)) → circ(cons(lift, circ(s, t)), u)
subst(a, id) → a
msubst(a, id) → a
msubst(msubst(a, s), t) → msubst(a, circ(s, t))

Types:
circ :: cons:id → cons:id → cons:id
cons :: lift → cons:id → cons:id
msubst :: lift → cons:id → lift
lift :: lift
id :: cons:id
subst :: subst → cons:id → subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat → cons:id

Lemmas:
circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0)) → gen_cons:id4_0(n16_0), rt ∈ Ω(1 + n16₀)

Generator Equations:
gen_cons:id4_0(0) ⇔ id
gen_cons:id4_0(+(x, 1)) ⇔ cons(lift, gen_cons:id4_0(x))

The following defined symbols remain to be analysed:
msubst

They will be analysed ascendingly in the following order:
circ = msubst

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

TRS:
Rules:
circ(cons(a, s), t) → cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) → cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) → cons(lift, circ(s, t))
circ(circ(s, t), u) → circ(s, circ(t, u))
circ(s, id) → s
circ(id, s) → s
circ(cons(lift, s), circ(cons(lift, t), u)) → circ(cons(lift, circ(s, t)), u)
subst(a, id) → a
msubst(a, id) → a
msubst(msubst(a, s), t) → msubst(a, circ(s, t))

Types:
circ :: cons:id → cons:id → cons:id
cons :: lift → cons:id → cons:id
msubst :: lift → cons:id → lift
lift :: lift
id :: cons:id
subst :: subst → cons:id → subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat → cons:id

Lemmas:
circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0)) → gen_cons:id4_0(n16_0), rt ∈ Ω(1 + n16₀)

Generator Equations:
gen_cons:id4_0(0) ⇔ id
gen_cons:id4_0(+(x, 1)) ⇔ cons(lift, gen_cons:id4_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0)) → gen_cons:id4_0(n16_0), rt ∈ Ω(1 + n16₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
circ(cons(a, s), t) → cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) → cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) → cons(lift, circ(s, t))
circ(circ(s, t), u) → circ(s, circ(t, u))
circ(s, id) → s
circ(id, s) → s
circ(cons(lift, s), circ(cons(lift, t), u)) → circ(cons(lift, circ(s, t)), u)
subst(a, id) → a
msubst(a, id) → a
msubst(msubst(a, s), t) → msubst(a, circ(s, t))

Types:
circ :: cons:id → cons:id → cons:id
cons :: lift → cons:id → cons:id
msubst :: lift → cons:id → lift
lift :: lift
id :: cons:id
subst :: subst → cons:id → subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat → cons:id

Lemmas:
circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0)) → gen_cons:id4_0(n16_0), rt ∈ Ω(1 + n16₀)

Generator Equations:
gen_cons:id4_0(0) ⇔ id
gen_cons:id4_0(+(x, 1)) ⇔ cons(lift, gen_cons:id4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0)) → gen_cons:id4_0(n16_0), rt ∈ Ω(1 + n16₀)

(20) BOUNDS(n^1, INF)