WORST_CASE(?,O(n^1))
proof of input_D37WvPia3s.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTRS
(11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(12) CpxTRS
(13) CpxTrsMatchBoundsTAProof [FINISHED, 25 ms]
(14) BOUNDS(1, n^1)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__eq(0, 0) -> true
   a__eq(s(z0), s(z1)) -> a__eq(z0, z1)
   a__eq(z0, z1) -> false
   a__eq(z0, z1) -> eq(z0, z1)
   a__inf(z0) -> cons(z0, inf(s(z0)))
   a__inf(z0) -> inf(z0)
   a__take(0, z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> s(length(z1))
   a__length(z0) -> length(z0)
   mark(eq(z0, z1)) -> a__eq(z0, z1)
   mark(inf(z0)) -> a__inf(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(length(z0)) -> a__length(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(s(z0)) -> s(z0)
   mark(false) -> false
   mark(cons(z0, z1)) -> cons(z0, z1)
   mark(nil) -> nil
Tuples:
   A__EQ(0, 0) -> c
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   A__EQ(z0, z1) -> c2
   A__EQ(z0, z1) -> c3
   A__INF(z0) -> c4
   A__INF(z0) -> c5
   A__TAKE(0, z0) -> c6
   A__TAKE(s(z0), cons(z1, z2)) -> c7
   A__TAKE(z0, z1) -> c8
   A__LENGTH(nil) -> c9
   A__LENGTH(cons(z0, z1)) -> c10
   A__LENGTH(z0) -> c11
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(A__INF(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c15(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(length(z0)) -> c16(A__LENGTH(mark(z0)), MARK(z0))
   MARK(0) -> c17
   MARK(true) -> c18
   MARK(s(z0)) -> c19
   MARK(false) -> c20
   MARK(cons(z0, z1)) -> c21
   MARK(nil) -> c22
S tuples:
   A__EQ(0, 0) -> c
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   A__EQ(z0, z1) -> c2
   A__EQ(z0, z1) -> c3
   A__INF(z0) -> c4
   A__INF(z0) -> c5
   A__TAKE(0, z0) -> c6
   A__TAKE(s(z0), cons(z1, z2)) -> c7
   A__TAKE(z0, z1) -> c8
   A__LENGTH(nil) -> c9
   A__LENGTH(cons(z0, z1)) -> c10
   A__LENGTH(z0) -> c11
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(A__INF(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c15(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(length(z0)) -> c16(A__LENGTH(mark(z0)), MARK(z0))
   MARK(0) -> c17
   MARK(true) -> c18
   MARK(s(z0)) -> c19
   MARK(false) -> c20
   MARK(cons(z0, z1)) -> c21
   MARK(nil) -> c22
K tuples:none
Defined Rule Symbols:   a__eq_2, a__inf_1, a__take_2, a__length_1, mark_1

Defined Pair Symbols:   A__EQ_2, A__INF_1, A__TAKE_2, A__LENGTH_1, MARK_1

Compound Symbols:   c, c1_1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12_1, c13_2, c14_2, c15_2, c16_2, c17, c18, c19, c20, c21, c22


----------------------------------------

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 17 trailing nodes:
   MARK(nil) -> c22
   A__TAKE(s(z0), cons(z1, z2)) -> c7
   A__INF(z0) -> c5
   MARK(false) -> c20
   A__EQ(z0, z1) -> c2
   A__LENGTH(nil) -> c9
   MARK(true) -> c18
   A__LENGTH(cons(z0, z1)) -> c10
   A__TAKE(0, z0) -> c6
   A__EQ(z0, z1) -> c3
   A__EQ(0, 0) -> c
   A__LENGTH(z0) -> c11
   MARK(cons(z0, z1)) -> c21
   A__INF(z0) -> c4
   MARK(0) -> c17
   A__TAKE(z0, z1) -> c8
   MARK(s(z0)) -> c19

----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__eq(0, 0) -> true
   a__eq(s(z0), s(z1)) -> a__eq(z0, z1)
   a__eq(z0, z1) -> false
   a__eq(z0, z1) -> eq(z0, z1)
   a__inf(z0) -> cons(z0, inf(s(z0)))
   a__inf(z0) -> inf(z0)
   a__take(0, z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> s(length(z1))
   a__length(z0) -> length(z0)
   mark(eq(z0, z1)) -> a__eq(z0, z1)
   mark(inf(z0)) -> a__inf(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(length(z0)) -> a__length(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(s(z0)) -> s(z0)
   mark(false) -> false
   mark(cons(z0, z1)) -> cons(z0, z1)
   mark(nil) -> nil
Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(A__INF(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c15(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(length(z0)) -> c16(A__LENGTH(mark(z0)), MARK(z0))
S tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(A__INF(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c15(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(length(z0)) -> c16(A__LENGTH(mark(z0)), MARK(z0))
K tuples:none
Defined Rule Symbols:   a__eq_2, a__inf_1, a__take_2, a__length_1, mark_1

Defined Pair Symbols:   A__EQ_2, MARK_1

Compound Symbols:   c1_1, c12_1, c13_2, c14_2, c15_2, c16_2


----------------------------------------

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__eq(0, 0) -> true
   a__eq(s(z0), s(z1)) -> a__eq(z0, z1)
   a__eq(z0, z1) -> false
   a__eq(z0, z1) -> eq(z0, z1)
   a__inf(z0) -> cons(z0, inf(s(z0)))
   a__inf(z0) -> inf(z0)
   a__take(0, z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> s(length(z1))
   a__length(z0) -> length(z0)
   mark(eq(z0, z1)) -> a__eq(z0, z1)
   mark(inf(z0)) -> a__inf(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(length(z0)) -> a__length(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(s(z0)) -> s(z0)
   mark(false) -> false
   mark(cons(z0, z1)) -> cons(z0, z1)
   mark(nil) -> nil
Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0))
   MARK(take(z0, z1)) -> c15(MARK(z1))
   MARK(length(z0)) -> c16(MARK(z0))
S tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0))
   MARK(take(z0, z1)) -> c15(MARK(z1))
   MARK(length(z0)) -> c16(MARK(z0))
K tuples:none
Defined Rule Symbols:   a__eq_2, a__inf_1, a__take_2, a__length_1, mark_1

Defined Pair Symbols:   A__EQ_2, MARK_1

Compound Symbols:   c1_1, c12_1, c13_1, c14_1, c15_1, c16_1


----------------------------------------

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   a__eq(0, 0) -> true
   a__eq(s(z0), s(z1)) -> a__eq(z0, z1)
   a__eq(z0, z1) -> false
   a__eq(z0, z1) -> eq(z0, z1)
   a__inf(z0) -> cons(z0, inf(s(z0)))
   a__inf(z0) -> inf(z0)
   a__take(0, z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> s(length(z1))
   a__length(z0) -> length(z0)
   mark(eq(z0, z1)) -> a__eq(z0, z1)
   mark(inf(z0)) -> a__inf(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(length(z0)) -> a__length(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(s(z0)) -> s(z0)
   mark(false) -> false
   mark(cons(z0, z1)) -> cons(z0, z1)
   mark(nil) -> nil

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0))
   MARK(take(z0, z1)) -> c15(MARK(z1))
   MARK(length(z0)) -> c16(MARK(z0))
S tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0))
   MARK(take(z0, z1)) -> c15(MARK(z1))
   MARK(length(z0)) -> c16(MARK(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   A__EQ_2, MARK_1

Compound Symbols:   c1_1, c12_1, c13_1, c14_1, c15_1, c16_1


----------------------------------------

(9) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0))
   MARK(take(z0, z1)) -> c15(MARK(z1))
   MARK(length(z0)) -> c16(MARK(z0))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(11) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0))
   MARK(take(z0, z1)) -> c15(MARK(z1))
   MARK(length(z0)) -> c16(MARK(z0))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
s0(0) -> 0
c10(0) -> 0
eq0(0, 0) -> 0
c120(0) -> 0
inf0(0) -> 0
c130(0) -> 0
take0(0, 0) -> 0
c140(0) -> 0
c150(0) -> 0
length0(0) -> 0
c160(0) -> 0
A__EQ0(0, 0) -> 1
MARK0(0) -> 2
A__EQ1(0, 0) -> 3
c11(3) -> 1
A__EQ1(0, 0) -> 4
c121(4) -> 2
MARK1(0) -> 5
c131(5) -> 2
MARK1(0) -> 6
c141(6) -> 2
MARK1(0) -> 7
c151(7) -> 2
MARK1(0) -> 8
c161(8) -> 2
c11(3) -> 3
c11(3) -> 4
c121(4) -> 5
c121(4) -> 6
c121(4) -> 7
c121(4) -> 8
c131(5) -> 5
c131(5) -> 6
c131(5) -> 7
c131(5) -> 8
c141(6) -> 5
c141(6) -> 6
c141(6) -> 7
c141(6) -> 8
c151(7) -> 5
c151(7) -> 6
c151(7) -> 7
c151(7) -> 8
c161(8) -> 5
c161(8) -> 6
c161(8) -> 7
c161(8) -> 8

----------------------------------------

(14)
BOUNDS(1, n^1)