WORST_CASE(?,O(n^1))
proof of input_oAgW3ogEny.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# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTRS
(11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(12) CpxTRS
(13) CpxTrsMatchBoundsProof [FINISHED, 9 ms]
(14) 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:

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

   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))

S is empty.
Rewrite Strategy: INNERMOST
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(11) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(12)
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:

   F(s(z0)) -> c2(F(z0))
   F(s(z0)) -> c3(F(z0))
   F(s(z0)) -> c1(F(z0))

S is empty.
Rewrite Strategy: INNERMOST
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(13) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[44, 45, 48, 49, 50]
{(44,45,[F_1|0]), (44,48,[c2_1|1]), (44,49,[c3_1|1]), (44,50,[c1_1|1]), (45,45,[s_1|0, c2_1|0, c3_1|0, c1_1|0]), (48,45,[F_1|1]), (48,48,[c2_1|1]), (48,49,[c3_1|1]), (48,50,[c1_1|1]), (49,45,[F_1|1]), (49,48,[c2_1|1]), (49,49,[c3_1|1]), (49,50,[c1_1|1]), (50,45,[F_1|1]), (50,48,[c2_1|1]), (50,49,[c3_1|1]), (50,50,[c1_1|1])}"
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(14)
BOUNDS(1, n^1)