WORST_CASE(?,O(n^1))
proof of input_uPM0UBRDzV.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 64 ms]
(8) CdtProblem
(9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) BOUNDS(1, 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) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
We considered the (Usable) Rules:none
And the 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))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*'(x_1, x_2)) = x_1 + x_2
   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1

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(8)
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:none
K 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))
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1


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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(10)
BOUNDS(1, 1)