Runtime Complexity (full) proof of /tmp/tmpFlGW3J/MYNAT_nosorts_noand

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 692 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), 759 ms)

    ↳10 typed CpxTrs

    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 262 ms)

    ↳12 typed CpxTrs

    ↳13 RewriteLemmaProof (LOWER BOUND(ID), 138 ms)

    ↳14 BEST

        ↳15 typed CpxTrs

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

            ↳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:

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
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(U22(tt, X2, X3)) →⁺ a__plus(a__x(mark(X3), mark(X2)), mark(X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X3 / U22(tt, X2, X3)].
The result substitution is [ ].

The rewrite sequence
mark(U22(tt, X2, X3)) →⁺ a__plus(a__x(mark(X3), mark(X2)), mark(X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [X3 / U22(tt, X2, X3)].
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, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0') → 0'
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
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__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0') → 0'
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Types:
a__U11 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
tt :: tt:s:0':U11:U12:plus:U21:U22:x
a__U12 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
s :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
a__plus :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
mark :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
a__U21 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
a__U22 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
a__x :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
0' :: tt:s:0':U11:U12:plus:U21:U22:x
U11 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
U12 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
plus :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
U21 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
U22 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
x :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
hole_tt:s:0':U11:U12:plus:U21:U22:x1_0 :: tt:s:0':U11:U12:plus:U21:U22:x
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0 :: Nat → tt:s:0':U11:U12:plus:U21:U22:x

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

Heuristically decided to analyse the following defined symbols:
a__U11, a__U12, a__plus, mark, a__U21, a__U22, a__x

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U11 = a__U21
a__U11 = a__U22
a__U11 = a__x
a__U12 = a__plus
a__U12 = mark
a__U12 = a__U21
a__U12 = a__U22
a__U12 = a__x
a__plus = mark
a__plus = a__U21
a__plus = a__U22
a__plus = a__x
mark = a__U21
mark = a__U22
mark = a__x
a__U21 = a__U22
a__U21 = a__x
a__U22 = a__x

(8) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U12, a__U11, a__plus, mark, a__U21, a__U22, a__x

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

Could not prove a rewrite lemma for the defined symbol a__U12.

(10) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__plus, a__U11, mark, a__U21, a__U22, a__x

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

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

(12) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11, a__U21, a__U22, a__x

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Induction Base:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n85291_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0))) →_IH
s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c85292_0))

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

(14) Complex Obligation (BEST)

(15) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U12, a__plus, a__U21, a__U22, a__x

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

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

(17) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U21, a__U12, a__plus, a__U22, a__x

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

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

(19) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U22, a__U12, a__plus, a__x

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

Could not prove a rewrite lemma for the defined symbol a__U22.

(21) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__x, a__U12, a__plus

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

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

(23) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U12, a__plus

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

Could not prove a rewrite lemma for the defined symbol a__U12.

(25) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__plus

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

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

(27) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

(29) BOUNDS(n^1, INF)

(30) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85291_0), rt ∈ Ω(1 + n85291₀)

(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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(12) Obligation:

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(14) Complex Obligation (BEST)

(15) Obligation:

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

(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)