Runtime Complexity (full) proof of /tmp/tmp5TxxZ9/wst99.xml

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 1291 ms)

↳2 BOUNDS(n^1, INF)

↳3 RenamingProof (⇔, 0 ms)

↳4 CpxRelTRS

    ↳5 SlicingProof (LOWER BOUND(ID), 0 ms)

    ↳6 CpxRelTRS

    ↳7 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳8 typed CpxTrs

    ↳9 OrderProof (LOWER BOUND(ID), 0 ms)

    ↳10 typed CpxTrs

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

    ↳12 typed CpxTrs

    ↳13 RewriteLemmaProof (LOWER BOUND(ID), 207 ms)

    ↳14 BEST

        ↳15 typed CpxTrs

            ↳16 NoRewriteLemmaProof (LOWER BOUND(ID), 1 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:

din(der(plus(X, Y))) → u21(din(der(X)), X, Y)
u21(dout(DX), X, Y) → u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) → dout(plus(DX, DY))
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) → u41(din(der(X)), X)
u41(dout(DX), X) → u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) → dout(DDX)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
din(der(plus(X, Y))) →⁺ u21(din(der(X)), X, Y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / plus(X, Y)].
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:

din(der(plus(X, Y))) → u21(din(der(X)), X, Y)
u21(dout(DX), X, Y) → u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) → dout(plus(DX, DY))
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) → u41(din(der(X)), X)
u41(dout(DX), X) → u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) → dout(DDX)

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
u21/1
u22/1
u22/2
u41/1
u42/1
u42/2

(6) Obligation:

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

din(der(plus(X, Y))) → u21(din(der(X)), Y)
u21(dout(DX), Y) → u22(din(der(Y)), DX)
u22(dout(DY), DX) → dout(plus(DX, DY))
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) → u41(din(der(X)))
u41(dout(DX)) → u42(din(der(DX)))
u42(dout(DDX)) → dout(DDX)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
din(der(plus(X, Y))) → u21(din(der(X)), Y)
u21(dout(DX), Y) → u22(din(der(Y)), DX)
u22(dout(DY), DX) → dout(plus(DX, DY))
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) → u41(din(der(X)))
u41(dout(DX)) → u42(din(der(DX)))
u42(dout(DDX)) → dout(DDX)

Types:
din :: plus:der:times → dout
der :: plus:der:times → plus:der:times
plus :: plus:der:times → plus:der:times → plus:der:times
u21 :: dout → plus:der:times → dout
dout :: plus:der:times → dout
u22 :: dout → plus:der:times → dout
times :: plus:der:times → plus:der:times → plus:der:times
u31 :: dout → plus:der:times → plus:der:times → dout
u32 :: dout → plus:der:times → plus:der:times → plus:der:times → dout
u41 :: dout → dout
u42 :: dout → dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat → plus:der:times

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
din, u41

They will be analysed ascendingly in the following order:
din = u41

(10) Obligation:

Generator Equations:
gen_plus:der:times3_0(0) ⇔ hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) ⇔ der(gen_plus:der:times3_0(x))

The following defined symbols remain to be analysed:
u41, din

They will be analysed ascendingly in the following order:
din = u41

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

Could not prove a rewrite lemma for the defined symbol u41.

(12) Obligation:

Generator Equations:
gen_plus:der:times3_0(0) ⇔ hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) ⇔ der(gen_plus:der:times3_0(x))

The following defined symbols remain to be analysed:
din

They will be analysed ascendingly in the following order:
din = u41

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

Proved the following rewrite lemma:
din(gen_plus:der:times3_0(+(2, n29_0))) → *4_0, rt ∈ Ω(n29₀)

Induction Base:
din(gen_plus:der:times3_0(+(2, 0)))

Induction Step:
din(gen_plus:der:times3_0(+(2, +(n29_0, 1)))) →_R^Ω(1)
u41(din(der(gen_plus:der:times3_0(+(1, n29_0))))) →_IH
u41(*4_0)

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

(14) Complex Obligation (BEST)

(15) Obligation:

Lemmas:
din(gen_plus:der:times3_0(+(2, n29_0))) → *4_0, rt ∈ Ω(n29₀)

Generator Equations:
gen_plus:der:times3_0(0) ⇔ hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) ⇔ der(gen_plus:der:times3_0(x))

The following defined symbols remain to be analysed:
u41

They will be analysed ascendingly in the following order:
din = u41

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

Could not prove a rewrite lemma for the defined symbol u41.

(17) Obligation:

Lemmas:
din(gen_plus:der:times3_0(+(2, n29_0))) → *4_0, rt ∈ Ω(n29₀)

Generator Equations:
gen_plus:der:times3_0(0) ⇔ hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) ⇔ der(gen_plus:der:times3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
din(gen_plus:der:times3_0(+(2, n29_0))) → *4_0, rt ∈ Ω(n29₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Lemmas:
din(gen_plus:der:times3_0(+(2, n29_0))) → *4_0, rt ∈ Ω(n29₀)

Generator Equations:
gen_plus:der:times3_0(0) ⇔ hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) ⇔ der(gen_plus:der:times3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
din(gen_plus:der:times3_0(+(2, n29_0))) → *4_0, rt ∈ Ω(n29₀)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

(4) Obligation:

(5) SlicingProof (LOWER BOUND(ID) transformation)

(6) Obligation:

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

(8) Obligation:

(9) OrderProof (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) LowerBoundsProof (EQUIVALENT transformation)

(19) BOUNDS(n^1, INF)

(20) Obligation:

(21) LowerBoundsProof (EQUIVALENT transformation)

(22) BOUNDS(n^1, INF)