WORST_CASE(?,O(n^1))
proof of input_fz2nioVhTV.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
(9) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(10) CpxTRS
(11) CpxTrsMatchBoundsTAProof [FINISHED, 12 ms]
(12) BOUNDS(1, n^1)


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(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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   FIRST(0, z0) -> c
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   FIRST(z0, z1) -> c2
   FROM(z0) -> c3
   FROM(z0) -> c4
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
   ACTIVATE(n__from(z0)) -> c6(FROM(z0))
   ACTIVATE(z0) -> c7
S tuples:
   FIRST(0, z0) -> c
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   FIRST(z0, z1) -> c2
   FROM(z0) -> c3
   FROM(z0) -> c4
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
   ACTIVATE(n__from(z0)) -> c6(FROM(z0))
   ACTIVATE(z0) -> c7
K tuples:none
Defined Rule Symbols:   first_2, from_1, activate_1

Defined Pair Symbols:   FIRST_2, FROM_1, ACTIVATE_1

Compound Symbols:   c, c1_1, c2, c3, c4, c5_1, c6_1, c7


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   ACTIVATE(z0) -> c7
   FROM(z0) -> c3
   FIRST(0, z0) -> c
   FROM(z0) -> c4
   FIRST(z0, z1) -> c2
   ACTIVATE(n__from(z0)) -> c6(FROM(z0))

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
S tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:   first_2, from_1, activate_1

Defined Pair Symbols:   FIRST_2, ACTIVATE_1

Compound Symbols:   c1_1, c5_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
S tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FIRST_2, ACTIVATE_1

Compound Symbols:   c1_1, c5_1


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

(8)
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:

   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))

S is empty.
Rewrite Strategy: INNERMOST
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(9) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(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:

   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))

S is empty.
Rewrite Strategy: INNERMOST
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(11) 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
cons0(0, 0) -> 0
c10(0) -> 0
n__first0(0, 0) -> 0
c50(0) -> 0
FIRST0(0, 0) -> 1
ACTIVATE0(0) -> 2
ACTIVATE1(0) -> 3
c11(3) -> 1
FIRST1(0, 0) -> 4
c51(4) -> 2
c11(3) -> 4
c51(4) -> 3

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(12)
BOUNDS(1, n^1)