Runtime Complexity (full) proof of /tmp/tmpgMch3R/MYNAT_nokinds

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 591 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), 135 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), 811 ms)

    ↳16 BEST

        ↳17 typed CpxTrs

            ↳18 NoRewriteLemmaProof (LOWER BOUND(ID), 0 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 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^1, INF)

        ↳30 typed CpxTrs

            ↳31 LowerBoundsProof (⇔, 0 ms)

            ↳32 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)))
U31(tt) → 0
U41(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(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)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
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
isNat(n__plus(n__isNat(X22196_3), V2)) →⁺ and(isNat(isNat(X22196_3)), n__isNat(activate(V2)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X22196_3 / n__plus(n__isNat(X22196_3), V2)].
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, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0'
U41(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0') → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0') → U31(isNat(N))
x(N, s(M)) → U41(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)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
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)))
U31(tt) → 0'
U41(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0') → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0') → U31(isNat(N))
x(N, s(M)) → U41(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)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
tt :: tt:n__0:n__plus:n__isNat:n__s:n__x
activate :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
U21 :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
s :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
plus :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
U31 :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
0' :: tt:n__0:n__plus:n__isNat:n__s:n__x
U41 :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
x :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
and :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
isNat :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
n__0 :: tt:n__0:n__plus:n__isNat:n__s:n__x
n__plus :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
n__isNat :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
n__s :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
n__x :: tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x → tt:n__0:n__plus:n__isNat:n__s:n__x
hole_tt:n__0:n__plus:n__isNat:n__s:n__x1_3 :: tt:n__0:n__plus:n__isNat:n__s:n__x
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3 :: Nat → tt:n__0:n__plus:n__isNat:n__s:n__x

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

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

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

(8) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

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

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

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

(10) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

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

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

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

(12) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

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

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

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

(14) Obligation:

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

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

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

Proved the following rewrite lemma:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

Induction Base:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, 0)))

Induction Step:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, +(n361_3, 1)))) →_R^Ω(1)
and(isNat(activate(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3)))), n__isNat(activate(tt))) →_R^Ω(1)
and(isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))), n__isNat(activate(tt))) →_IH
and(*3_3, n__isNat(activate(tt))) →_R^Ω(1)
and(*3_3, n__isNat(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_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

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

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

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

(19) Obligation:

Lemmas:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

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

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

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

(21) Obligation:

Lemmas:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

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

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

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

(23) Obligation:

Lemmas:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

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

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

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

(25) Obligation:

Lemmas:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

The following defined symbols remain to be analysed:
U11

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

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

(27) Obligation:

Lemmas:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

(29) BOUNDS(n^1, INF)

(30) Obligation:

Lemmas:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

Generator Equations:
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(0) ⇔ tt
gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(x, 1)) ⇔ n__plus(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(x), tt)

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isNat(gen_tt:n__0:n__plus:n__isNat:n__s:n__x2_3(+(1, n361_3))) → *3_3, rt ∈ Ω(n361₃)

(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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(25) Obligation:

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

(27) Obligation:

(28) LowerBoundsProof (EQUIVALENT transformation)

(29) BOUNDS(n^1, INF)

(30) Obligation:

(31) LowerBoundsProof (EQUIVALENT transformation)

(32) BOUNDS(n^1, INF)