WORST_CASE(?,O(n^1))
proof of input_TvN0N0k2aK.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms]
(10) CdtProblem
(11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) 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:

   f(0) -> 1
   f(s(x)) -> g(f(x))
   g(x) -> +(x, s(x))
   f(s(x)) -> +(f(x), s(f(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:
   f(0) -> 1
   f(s(z0)) -> g(f(z0))
   f(s(z0)) -> +(f(z0), s(f(z0)))
   g(z0) -> +(z0, s(z0))
Tuples:
   F(0) -> c
   F(s(z0)) -> c1(G(f(z0)), F(z0))
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   G(z0) -> c4
S tuples:
   F(0) -> c
   F(s(z0)) -> c1(G(f(z0)), F(z0))
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   G(z0) -> c4
K tuples:none
Defined Rule Symbols:   f_1, g_1

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c, c1_2, c2_1, c3_1, c4


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   G(z0) -> c4
   F(0) -> c

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> 1
   f(s(z0)) -> g(f(z0))
   f(s(z0)) -> +(f(z0), s(f(z0)))
   g(z0) -> +(z0, s(z0))
Tuples:
   F(s(z0)) -> c1(G(f(z0)), F(z0))
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
S tuples:
   F(s(z0)) -> c1(G(f(z0)), F(z0))
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
K tuples:none
Defined Rule Symbols:   f_1, g_1

Defined Pair Symbols:   F_1

Compound Symbols:   c1_2, c2_1, c3_1


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> 1
   f(s(z0)) -> g(f(z0))
   f(s(z0)) -> +(f(z0), s(f(z0)))
   g(z0) -> +(z0, s(z0))
Tuples:
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))
S tuples:
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))
K tuples:none
Defined Rule Symbols:   f_1, g_1

Defined Pair Symbols:   F_1

Compound Symbols:   c2_1, c3_1, c1_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0) -> 1
   f(s(z0)) -> g(f(z0))
   f(s(z0)) -> +(f(z0), s(f(z0)))
   g(z0) -> +(z0, s(z0))

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))
S tuples:
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1

Compound Symbols:   c2_1, c3_1, c1_1


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))
S tuples:none
K tuples:
   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1

Compound Symbols:   c2_1, c3_1, c1_1


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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(12)
BOUNDS(1, 1)