Runtime Complexity (full) proof of /tmp/tmpgFfzCS/PEANO_nokinds

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 190 ms)

↳2 BOUNDS(2^n, INF)

↳3 RenamingProof (⇔, 0 ms)

↳4 CpxRelTRS

    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 4 ms)

    ↳6 typed CpxTrs

    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)

    ↳8 typed CpxTrs

    ↳9 RewriteLemmaProof (LOWER BOUND(ID), 379 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

            ↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

            ↳13 typed CpxTrs

            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

            ↳15 typed CpxTrs

            ↳16 RewriteLemmaProof (LOWER BOUND(ID), 132 ms)

            ↳17 BEST

                ↳18 typed CpxTrs

                    ↳19 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

                    ↳20 typed CpxTrs

                    ↳21 RewriteLemmaProof (LOWER BOUND(ID), 43 ms)

                    ↳22 BEST

                        ↳23 typed CpxTrs

                            ↳24 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

                            ↳25 typed CpxTrs

                            ↳26 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

                            ↳27 typed CpxTrs

                            ↳28 LowerBoundsProof (⇔, 0 ms)

                            ↳29 BOUNDS(n^2, INF)

                        ↳30 typed CpxTrs

                            ↳31 LowerBoundsProof (⇔, 0 ms)

                            ↳32 BOUNDS(n^2, INF)

                ↳33 typed CpxTrs

                    ↳34 LowerBoundsProof (⇔, 0 ms)

                    ↳35 BOUNDS(n^2, INF)

        ↳36 typed CpxTrs

            ↳37 LowerBoundsProof (⇔, 0 ms)

            ↳38 BOUNDS(n^1, INF)

(0) Obligation:

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

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
activate(n__plus(X1, n__0)) →⁺ U11(isNat(activate(X1)), activate(X1))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X1 / n__plus(X1, n__0)].
The result substitution is [ ].

The rewrite sequence
activate(n__plus(X1, n__0)) →⁺ U11(isNat(activate(X1)), activate(X1))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [X1 / n__plus(X1, n__0)].
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:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0') → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0') → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
U11 :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
tt :: tt:n__0:n__plus:n__isNat:n__s
activate :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
U21 :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
s :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
plus :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
and :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
isNat :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
n__0 :: tt:n__0:n__plus:n__isNat:n__s
n__plus :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
n__isNat :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
n__s :: tt:n__0:n__plus:n__isNat:n__s → tt:n__0:n__plus:n__isNat:n__s
0' :: tt:n__0:n__plus:n__isNat:n__s
hole_tt:n__0:n__plus:n__isNat:n__s1_3 :: tt:n__0:n__plus:n__isNat:n__s
gen_tt:n__0:n__plus:n__isNat:n__s2_3 :: Nat → tt:n__0:n__plus:n__isNat:n__s

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

Heuristically decided to analyse the following defined symbols:
U11, activate, plus, and, isNat

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

(8) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

The following defined symbols remain to be analysed:
activate, U11, plus, and, isNat

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

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

Proved the following rewrite lemma:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), rt ∈ Ω(1 + n4₃)

Induction Base:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(0)) →_R^Ω(1)
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0)

Induction Step:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(n4_3, 1))) →_R^Ω(1)
plus(activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3)), activate(n__0)) →_IH
plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(c5_3), activate(n__0)) →_R^Ω(1)
plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), n__0) →_R^Ω(1)
n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), n__0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Lemmas:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), rt ∈ Ω(1 + n4₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

The following defined symbols remain to be analysed:
plus, U11, and, isNat

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

Lemmas:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), rt ∈ Ω(1 + n4₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

The following defined symbols remain to be analysed:
U11, and, isNat

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

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

Could not prove a rewrite lemma for the defined symbol U11.

(15) Obligation:

Lemmas:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), rt ∈ Ω(1 + n4₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

The following defined symbols remain to be analysed:
isNat, and

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n3258_3)) → tt, rt ∈ Ω(1 + n3258₃ + n3258₃²)

Induction Base:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(0)) →_R^Ω(1)
tt

Induction Step:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(n3258_3, 1))) →_R^Ω(1)
and(isNat(activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n3258_3))), n__isNat(activate(n__0))) →_L^{Ω(1 + n3258₃)}
and(isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n3258_3)), n__isNat(activate(n__0))) →_IH
and(tt, n__isNat(activate(n__0))) →_L^Ω(1)
and(tt, n__isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(0))) →_R^Ω(1)
activate(n__isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(0))) →_R^Ω(1)
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(0)) →_R^Ω(1)
tt

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

(17) Complex Obligation (BEST)

(18) Obligation:

Lemmas:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), rt ∈ Ω(1 + n4₃)
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n3258_3)) → tt, rt ∈ Ω(1 + n3258₃ + n3258₃²)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

The following defined symbols remain to be analysed:
and, U11, activate, plus

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

The following defined symbols remain to be analysed:
activate, U11, plus

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n5192_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n5192_3), rt ∈ Ω(1 + n5192₃)

Induction Base:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(0)) →_R^Ω(1)
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0)

Induction Step:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(n5192_3, 1))) →_R^Ω(1)
plus(activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n5192_3)), activate(n__0)) →_IH
plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(c5193_3), activate(n__0)) →_R^Ω(1)
plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n5192_3), n__0) →_R^Ω(1)
n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n5192_3), n__0)

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

(22) Complex Obligation (BEST)

(23) Obligation:

Lemmas:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n5192_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n5192_3), rt ∈ Ω(1 + n5192₃)
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n3258_3)) → tt, rt ∈ Ω(1 + n3258₃ + n3258₃²)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

The following defined symbols remain to be analysed:
plus, U11

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(25) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

The following defined symbols remain to be analysed:
U11

They will be analysed ascendingly in the following order:
U11 = activate
U11 = plus
U11 = and
U11 = isNat
activate = plus
activate = and
activate = isNat
plus = and
plus = isNat
and = isNat

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U11.

(27) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n3258_3)) → tt, rt ∈ Ω(1 + n3258₃ + n3258₃²)

(29) BOUNDS(n^2, INF)

(30) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n3258_3)) → tt, rt ∈ Ω(1 + n3258₃ + n3258₃²)

(32) BOUNDS(n^2, INF)

(33) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n3258_3)) → tt, rt ∈ Ω(1 + n3258₃ + n3258₃²)

(35) BOUNDS(n^2, INF)

(36) Obligation:

Lemmas:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), rt ∈ Ω(1 + n4₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s2_3(0) ⇔ n__0
gen_tt:n__0:n__plus:n__isNat:n__s2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s2_3(x), n__0)

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3)) → gen_tt:n__0:n__plus:n__isNat:n__s2_3(n4_3), rt ∈ Ω(1 + n4₃)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(2^n, INF)

(3) RenamingProof (EQUIVALENT transformation)

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Complex Obligation (BEST)

(11) Obligation:

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(13) Obligation:

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

(15) Obligation:

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(17) Complex Obligation (BEST)

(18) Obligation:

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(20) Obligation:

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(22) Complex Obligation (BEST)

(23) Obligation:

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(25) Obligation:

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(27) Obligation:

(28) LowerBoundsProof (EQUIVALENT transformation)

(29) BOUNDS(n^2, INF)

(30) Obligation:

(31) LowerBoundsProof (EQUIVALENT transformation)

(32) BOUNDS(n^2, INF)

(33) Obligation:

(34) LowerBoundsProof (EQUIVALENT transformation)

(35) BOUNDS(n^2, INF)

(36) Obligation:

(37) LowerBoundsProof (EQUIVALENT transformation)

(38) BOUNDS(n^1, INF)