WORST_CASE(?,O(n^1))
proof of input_qxGlVthuyU.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
(4) 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:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(g(X)) -> g(X)
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
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:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(g(X)) -> g(X)
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(3) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. 
The certificate found is represented by the following graph.

"[6, 7, 8, 9, 10, 11, 12, 13, 14]
{(6,7,[a__f_1|0, mark_1|0, f_1|1, a|1, g_1|1]), (6,8,[a__f_1|1, f_1|2]), (6,11,[a__f_1|1, f_1|2]), (6,12,[a__f_1|2, f_1|3]), (7,7,[f_1|0, a|0, g_1|0]), (8,9,[g_1|1]), (9,10,[f_1|1]), (10,7,[a|1]), (11,7,[mark_1|1, a|1, g_1|1]), (11,11,[a__f_1|1, f_1|2]), (11,12,[a__f_1|2, f_1|3]), (12,13,[g_1|2]), (13,14,[f_1|2]), (14,7,[a|2])}"
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(4)
BOUNDS(1, n^1)