WORST_CASE(?,O(n^1))
proof of input_YZMzOHfLBB.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 15 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:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, 1) -> x
   *(1, y) -> y

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))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4, c5


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(1, z0) -> c5
   *'(z0, 1) -> c4

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1, c1_1, c2_1, c3_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.
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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))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))

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

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))

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]
transitions: 
+0(0, 0) -> 0
c0(0) -> 0
c10(0) -> 0
c20(0) -> 0
c30(0) -> 0
*'0(0, 0) -> 1
*'1(0, 0) -> 2
c1(2) -> 1
*'1(0, 0) -> 3
c11(3) -> 1
*'1(0, 0) -> 4
c21(4) -> 1
*'1(0, 0) -> 5
c31(5) -> 1
c1(2) -> 2
c1(2) -> 3
c1(2) -> 4
c1(2) -> 5
c11(3) -> 2
c11(3) -> 3
c11(3) -> 4
c11(3) -> 5
c21(4) -> 2
c21(4) -> 3
c21(4) -> 4
c21(4) -> 5
c31(5) -> 2
c31(5) -> 3
c31(5) -> 4
c31(5) -> 5

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(12)
BOUNDS(1, n^1)