The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 190 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 typed CpxTrs
    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 30 ms)
    ↳10 typed CpxTrs

(0) Obligation:

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

minus(minus(x)) → x
minus(h(x)) → h(minus(x))
minus(f(x, y)) → f(minus(y), minus(x))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
minus(h(x)) →+ h(minus(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / h(x)].
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:

minus(minus(x)) → x
minus(h(x)) → h(minus(x))
minus(f(x, y)) → f(minus(y), minus(x))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
minus(minus(x)) → x
minus(h(x)) → h(minus(x))
minus(f(x, y)) → f(minus(y), minus(x))

Types:
minus :: h:f → h:f
h :: h:f → h:f
f :: h:f → h:f → h:f
hole_h:f1_0 :: h:f
gen_h:f2_0 :: Nat → h:f

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

Heuristically decided to analyse the following defined symbols:
minus

(8) Obligation:

TRS:
Rules:
minus(minus(x)) → x
minus(h(x)) → h(minus(x))
minus(f(x, y)) → f(minus(y), minus(x))

Types:
minus :: h:f → h:f
h :: h:f → h:f
f :: h:f → h:f → h:f
hole_h:f1_0 :: h:f
gen_h:f2_0 :: Nat → h:f

Generator Equations:
gen_h:f2_0(0) ⇔ hole_h:f1_0
gen_h:f2_0(+(x, 1)) ⇔ h(gen_h:f2_0(x))

The following defined symbols remain to be analysed:
minus

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

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

(10) Obligation:

TRS:
Rules:
minus(minus(x)) → x
minus(h(x)) → h(minus(x))
minus(f(x, y)) → f(minus(y), minus(x))

Types:
minus :: h:f → h:f
h :: h:f → h:f
f :: h:f → h:f → h:f
hole_h:f1_0 :: h:f
gen_h:f2_0 :: Nat → h:f

Generator Equations:
gen_h:f2_0(0) ⇔ hole_h:f1_0
gen_h:f2_0(+(x, 1)) ⇔ h(gen_h:f2_0(x))

No more defined symbols left to analyse.