Runtime Complexity (full) proof of /tmp/tmpNSjWmd/MYNAT_nosorts

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 391 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 NoRewriteLemmaProof (LOWER BOUND(ID), 319 ms)

    ↳10 typed CpxTrs

    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 101 ms)

    ↳12 BEST

        ↳13 typed CpxTrs

            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 7 ms)

            ↳15 typed CpxTrs

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

            ↳17 typed CpxTrs

            ↳18 LowerBoundsProof (⇔, 0 ms)

            ↳19 BOUNDS(n^1, INF)

        ↳20 typed CpxTrs

            ↳21 LowerBoundsProof (⇔, 0 ms)

            ↳22 BOUNDS(n^1, INF)

(0) Obligation:

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

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0) → 0
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
mark(and(x(X110212_3, s(X10414_3)), X2)) →⁺ a__and(a__plus(a__x(mark(mark(X110212_3)), mark(mark(X10414_3))), mark(mark(X110212_3))), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0].
The pumping substitution is [X110212_3 / and(x(X110212_3, s(X10414_3)), X2)].
The result substitution is [ ].

The rewrite sequence
mark(and(x(X110212_3, s(X10414_3)), X2)) →⁺ a__and(a__plus(a__x(mark(mark(X110212_3)), mark(mark(X10414_3))), mark(mark(X110212_3))), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].
The pumping substitution is [X110212_3 / and(x(X110212_3, s(X10414_3)), 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__and(tt, X) → mark(X)
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0') → 0'
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__and(tt, X) → mark(X)
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0') → 0'
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

Types:
a__and :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
tt :: tt:0':s:and:plus:x
mark :: tt:0':s:and:plus:x → tt:0':s:and:plus:x
a__plus :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
0' :: tt:0':s:and:plus:x
s :: tt:0':s:and:plus:x → tt:0':s:and:plus:x
a__x :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
and :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
plus :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
x :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
hole_tt:0':s:and:plus:x1_0 :: tt:0':s:and:plus:x
gen_tt:0':s:and:plus:x2_0 :: Nat → tt:0':s:and:plus:x

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

Heuristically decided to analyse the following defined symbols:
mark, a__plus, a__x

They will be analysed ascendingly in the following order:
mark = a__plus
mark = a__x
a__plus = a__x

(8) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

They will be analysed ascendingly in the following order:
mark = a__plus
mark = a__x
a__plus = a__x

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

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

(10) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

The following defined symbols remain to be analysed:
mark, a__x

They will be analysed ascendingly in the following order:
mark = a__plus
mark = a__x
a__plus = a__x

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:0':s:and:plus:x2_0(n11388_0)) → gen_tt:0':s:and:plus:x2_0(n11388_0), rt ∈ Ω(1 + n11388₀)

Induction Base:
mark(gen_tt:0':s:and:plus:x2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:0':s:and:plus:x2_0(+(n11388_0, 1))) →_R^Ω(1)
s(mark(gen_tt:0':s:and:plus:x2_0(n11388_0))) →_IH
s(gen_tt:0':s:and:plus:x2_0(c11389_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11388_0)) → gen_tt:0':s:and:plus:x2_0(n11388_0), rt ∈ Ω(1 + n11388₀)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

They will be analysed ascendingly in the following order:
mark = a__plus
mark = a__x
a__plus = a__x

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

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

(15) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11388_0)) → gen_tt:0':s:and:plus:x2_0(n11388_0), rt ∈ Ω(1 + n11388₀)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

The following defined symbols remain to be analysed:
a__plus

They will be analysed ascendingly in the following order:
mark = a__plus
mark = a__x
a__plus = a__x

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

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

(17) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11388_0)) → gen_tt:0':s:and:plus:x2_0(n11388_0), rt ∈ Ω(1 + n11388₀)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:0':s:and:plus:x2_0(n11388_0)) → gen_tt:0':s:and:plus:x2_0(n11388_0), rt ∈ Ω(1 + n11388₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11388_0)) → gen_tt:0':s:and:plus:x2_0(n11388_0), rt ∈ Ω(1 + n11388₀)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:0':s:and:plus:x2_0(n11388_0)) → gen_tt:0':s:and:plus:x2_0(n11388_0), rt ∈ Ω(1 + n11388₀)

(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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Obligation:

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(12) Complex Obligation (BEST)

(13) Obligation:

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

(15) Obligation:

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

(17) Obligation:

(18) LowerBoundsProof (EQUIVALENT transformation)

(19) BOUNDS(n^1, INF)

(20) Obligation:

(21) LowerBoundsProof (EQUIVALENT transformation)

(22) BOUNDS(n^1, INF)