Runtime Complexity (full) proof of /tmp/tmpws5oV6/PEANO_nokinds-noand

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 489 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), 141 ms)

    ↳10 typed CpxTrs

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

    ↳12 typed CpxTrs

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

    ↳14 typed CpxTrs

    ↳15 RewriteLemmaProof (LOWER BOUND(ID), 58 ms)

    ↳16 BEST

        ↳17 typed CpxTrs

            ↳18 NoRewriteLemmaProof (LOWER BOUND(ID), 6 ms)

            ↳19 typed CpxTrs

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

            ↳21 typed CpxTrs

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

            ↳23 typed CpxTrs

            ↳24 LowerBoundsProof (⇔, 0 ms)

            ↳25 BOUNDS(n^1, INF)

        ↳26 typed CpxTrs

            ↳27 LowerBoundsProof (⇔, 0 ms)

            ↳28 BOUNDS(n^1, INF)

(0) Obligation:

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

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
U11(tt, n__plus(n__0, V2236_3)) →⁺ U12(U11(tt, V2236_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [V2236_3 / n__plus(n__0, V2236_3)].
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:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0') → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(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, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0') → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

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

Heuristically decided to analyse the following defined symbols:
isNat, activate, U31, plus

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(8) Obligation:

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
activate, isNat, U31, plus

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

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

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

(10) Obligation:

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
plus, isNat, U31

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

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

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

(12) Obligation:

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
U31, isNat

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

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

Could not prove a rewrite lemma for the defined symbol U31.

(14) Obligation:

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
isNat

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
isNat(gen_n__0:n__plus:n__s3_3(n155_3)) → tt, rt ∈ Ω(1 + n155₃)

Induction Base:
isNat(gen_n__0:n__plus:n__s3_3(0)) →_R^Ω(1)
tt

Induction Step:
isNat(gen_n__0:n__plus:n__s3_3(+(n155_3, 1))) →_R^Ω(1)
U11(isNat(activate(gen_n__0:n__plus:n__s3_3(n155_3))), activate(n__0)) →_R^Ω(1)
U11(isNat(gen_n__0:n__plus:n__s3_3(n155_3)), activate(n__0)) →_IH
U11(tt, activate(n__0)) →_R^Ω(1)
U11(tt, n__0) →_R^Ω(1)
U12(isNat(activate(n__0))) →_R^Ω(1)
U12(isNat(n__0)) →_R^Ω(1)
U12(tt) →_R^Ω(1)
tt

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

(16) Complex Obligation (BEST)

(17) Obligation:

Lemmas:
isNat(gen_n__0:n__plus:n__s3_3(n155_3)) → tt, rt ∈ Ω(1 + n155₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
activate, U31, plus

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

Lemmas:
isNat(gen_n__0:n__plus:n__s3_3(n155_3)) → tt, rt ∈ Ω(1 + n155₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
plus, U31

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(21) Obligation:

Lemmas:
isNat(gen_n__0:n__plus:n__s3_3(n155_3)) → tt, rt ∈ Ω(1 + n155₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
U31

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U31.

(23) Obligation:

Lemmas:
isNat(gen_n__0:n__plus:n__s3_3(n155_3)) → tt, rt ∈ Ω(1 + n155₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isNat(gen_n__0:n__plus:n__s3_3(n155_3)) → tt, rt ∈ Ω(1 + n155₃)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Lemmas:
isNat(gen_n__0:n__plus:n__s3_3(n155_3)) → tt, rt ∈ Ω(1 + n155₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isNat(gen_n__0:n__plus:n__s3_3(n155_3)) → tt, rt ∈ Ω(1 + n155₃)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(n^1, 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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(16) Complex Obligation (BEST)

(17) Obligation:

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(19) Obligation:

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(21) Obligation:

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(23) Obligation:

(24) LowerBoundsProof (EQUIVALENT transformation)

(25) BOUNDS(n^1, INF)

(26) Obligation:

(27) LowerBoundsProof (EQUIVALENT transformation)

(28) BOUNDS(n^1, INF)