WORST_CASE(?,O(n^1))
proof of input_nAJofCd8sM.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
(7) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(8) CpxTRS
(9) CpxTrsMatchBoundsTAProof [FINISHED, 16 ms]
(10) 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:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))

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:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1, c1_1


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(3) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1, c1_1


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(5) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
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(6)
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:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))

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

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))

S is empty.
Rewrite Strategy: INNERMOST
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(9) 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]
transitions: 
+0(0, 0) -> 0
c0(0) -> 0
c10(0) -> 0
*'0(0, 0) -> 1
*'1(0, 0) -> 2
c1(2) -> 1
*'1(0, 0) -> 3
c11(3) -> 1
c1(2) -> 2
c1(2) -> 3
c11(3) -> 2
c11(3) -> 3

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(10)
BOUNDS(1, n^1)