Runtime Complexity (full) proof of /tmp/tmplDiMp1/PEANO_nokinds

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 790 ms)

↳2 BOUNDS(2^n, 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 RewriteLemmaProof (LOWER BOUND(ID), 380 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

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

            ↳13 typed CpxTrs

            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 28 ms)

            ↳15 typed CpxTrs

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

            ↳17 typed CpxTrs

            ↳18 RewriteLemmaProof (LOWER BOUND(ID), 130 ms)

            ↳19 BEST

                ↳20 typed CpxTrs

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

                    ↳22 typed CpxTrs

                    ↳23 RewriteLemmaProof (LOWER BOUND(ID), 64 ms)

                    ↳24 BEST

                        ↳25 typed CpxTrs

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

                            ↳27 typed CpxTrs

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

                            ↳29 typed CpxTrs

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

                            ↳31 typed CpxTrs

                            ↳32 LowerBoundsProof (⇔, 0 ms)

                            ↳33 BOUNDS(n^1, INF)

                        ↳34 typed CpxTrs

                            ↳35 LowerBoundsProof (⇔, 0 ms)

                            ↳36 BOUNDS(n^1, INF)

                ↳37 typed CpxTrs

                    ↳38 LowerBoundsProof (⇔, 0 ms)

                    ↳39 BOUNDS(n^1, INF)

        ↳40 typed CpxTrs

            ↳41 LowerBoundsProof (⇔, 0 ms)

            ↳42 BOUNDS(n^1, INF)

(0) Obligation:

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

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
mark(U11(plus(X173362_4, s(X73646_4)), X2)) →⁺ a__U11(a__U21(a__and(a__isNat(mark(X73646_4)), isNat(mark(X173362_4))), mark(X73646_4), mark(X173362_4)), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0].
The pumping substitution is [X73646_4 / U11(plus(X173362_4, s(X73646_4)), X2)].
The result substitution is [ ].

The rewrite sequence
mark(U11(plus(X173362_4, s(X73646_4)), X2)) →⁺ a__U11(a__U21(a__and(a__isNat(mark(X73646_4)), isNat(mark(X173362_4))), mark(X73646_4), mark(X173362_4)), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
The pumping substitution is [X73646_4 / U11(plus(X173362_4, s(X73646_4)), X2)].
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:

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0') → tt
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0') → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0') → tt
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0') → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Types:
a__U11 :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
tt :: tt:s:0':plus:isNat:U11:U21:and
mark :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
a__U21 :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
s :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
a__plus :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
a__and :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
a__isNat :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
0' :: tt:s:0':plus:isNat:U11:U21:and
plus :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
isNat :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
U11 :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
U21 :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
and :: tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and → tt:s:0':plus:isNat:U11:U21:and
hole_tt:s:0':plus:isNat:U11:U21:and1_0 :: tt:s:0':plus:isNat:U11:U21:and
gen_tt:s:0':plus:isNat:U11:U21:and2_0 :: Nat → tt:s:0':plus:isNat:U11:U21:and

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

Heuristically decided to analyse the following defined symbols:
a__U11, mark, a__U21, a__plus, a__and, a__isNat

They will be analysed ascendingly in the following order:
a__U11 = mark
a__U11 = a__U21
a__U11 = a__plus
a__U11 = a__and
a__U11 = a__isNat
mark = a__U21
mark = a__plus
mark = a__and
mark = a__isNat
a__U21 = a__plus
a__U21 = a__and
a__U21 = a__isNat
a__plus = a__and
a__plus = a__isNat
a__and = a__isNat

(8) Obligation:

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11, a__U21, a__plus, a__and, a__isNat

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

Proved the following rewrite lemma:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(n4_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0))) →_IH
s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(c5_0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Lemmas:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U21, a__plus, a__and, a__isNat

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

Could not prove a rewrite lemma for the defined symbol a__U11.

(13) Obligation:

Lemmas:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
a__U21, a__plus, a__and, a__isNat

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

Could not prove a rewrite lemma for the defined symbol a__U21.

(15) Obligation:

Lemmas:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
a__plus, a__and, a__isNat

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Lemmas:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
a__isNat, a__and

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__isNat(gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(1, n4611_0))) → *3_0, rt ∈ Ω(n4611₀)

Induction Base:
a__isNat(gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(1, 0)))

Induction Step:
a__isNat(gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(1, +(n4611_0, 1)))) →_R^Ω(1)
a__isNat(gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(1, n4611_0))) →_IH
*3_0

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

(19) Complex Obligation (BEST)

(20) Obligation:

Lemmas:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)
a__isNat(gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(1, n4611_0))) → *3_0, rt ∈ Ω(n4611₀)

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
a__and, a__U11, mark, a__U21, a__plus

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11, a__U21, a__plus

(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0), rt ∈ Ω(1 + n5734₀)

Induction Base:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(n5734_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0))) →_IH
s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(c5735_0))

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

(24) Complex Obligation (BEST)

(25) Obligation:

Lemmas:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0), rt ∈ Ω(1 + n5734₀)
a__isNat(gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(1, n4611_0))) → *3_0, rt ∈ Ω(n4611₀)

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U21, a__plus

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

Could not prove a rewrite lemma for the defined symbol a__U11.

(27) Obligation:

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
a__U21, a__plus

(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__U21.

(29) Obligation:

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

The following defined symbols remain to be analysed:
a__plus

(30) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(31) Obligation:

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0), rt ∈ Ω(1 + n5734₀)

(33) BOUNDS(n^1, INF)

(34) Obligation:

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n5734_0), rt ∈ Ω(1 + n5734₀)

(36) BOUNDS(n^1, INF)

(37) Obligation:

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

No more defined symbols left to analyse.

(38) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)

(39) BOUNDS(n^1, INF)

(40) Obligation:

Lemmas:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_tt:s:0':plus:isNat:U11:U21:and2_0(0) ⇔ tt
gen_tt:s:0':plus:isNat:U11:U21:and2_0(+(x, 1)) ⇔ s(gen_tt:s:0':plus:isNat:U11:U21:and2_0(x))

No more defined symbols left to analyse.

(41) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0)) → gen_tt:s:0':plus:isNat:U11:U21:and2_0(n4_0), rt ∈ Ω(1 + n4₀)

(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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(17) Obligation:

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(19) Complex Obligation (BEST)

(20) Obligation:

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(22) Obligation:

(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(24) Complex Obligation (BEST)

(25) Obligation:

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

(27) Obligation:

(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(29) Obligation:

(30) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(31) Obligation:

(32) LowerBoundsProof (EQUIVALENT transformation)

(33) BOUNDS(n^1, INF)

(34) Obligation:

(35) LowerBoundsProof (EQUIVALENT transformation)

(36) BOUNDS(n^1, INF)

(37) Obligation:

(38) LowerBoundsProof (EQUIVALENT transformation)

(39) BOUNDS(n^1, INF)

(40) Obligation:

(41) LowerBoundsProof (EQUIVALENT transformation)

(42) BOUNDS(n^1, INF)