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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 201 ms]
(2) CpxRelTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsMatchBoundsTAProof [FINISHED, 25 ms]
(6) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   add0(S(x), x2) -> +(S(0), add0(x2, x))
   add0(0, x2) -> x2

The (relative) TRS S consists of the following rules:

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)

Rewrite Strategy: PARALLEL_INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   add0(S(x), x2) -> +(S(0), add0(x2, x))
   add0(0, x2) -> x2

The (relative) TRS S consists of the following rules:

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)

Rewrite Strategy: PARALLEL_INNERMOST
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
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:

   add0(S(x), x2) -> +(S(0), add0(x2, x))
   add0(0, x2) -> x2
   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(5) 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 2. 

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
00() -> 0
add00(0, 0) -> 1
+0(0, 0) -> 2
01() -> 4
S1(4) -> 3
add01(0, 0) -> 5
+1(3, 5) -> 1
S1(0) -> 2
+1(3, 5) -> 5
S2(5) -> 1
S1(3) -> 1
S1(3) -> 5
S2(5) -> 5
0 -> 1
0 -> 5

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(6)
BOUNDS(1, n^1)