WORST_CASE(?,O(n^1))
proof of input_ofpC8KDzhX.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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:

   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   g(X) -> n__g(X)
   activate(n__f(X)) -> f(X)
   activate(n__a) -> a
   activate(n__g(X)) -> g(activate(X))
   activate(X) -> 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:

   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   g(X) -> n__g(X)
   activate(n__f(X)) -> f(X)
   activate(n__a) -> a
   activate(n__g(X)) -> g(activate(X))
   activate(X) -> 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 2. 
The certificate found is represented by the following graph.

"[1, 2, 3, 4, 5, 6]
{(1,2,[f_1|0, a|0, g_1|0, activate_1|0, n__f_1|1, n__a|1, n__g_1|1, f_1|1, a|1, n__f_1|2, n__a|2]), (1,3,[f_1|1, n__f_1|2]), (1,6,[g_1|1, n__g_1|2]), (2,2,[n__f_1|0, n__a|0, n__g_1|0]), (3,4,[n__g_1|1]), (4,5,[n__f_1|1]), (5,2,[n__a|1]), (6,2,[activate_1|1, f_1|1, n__f_1|1, a|1, n__a|1, n__g_1|1, n__f_1|2, n__a|2]), (6,6,[g_1|1, n__g_1|2]), (6,3,[f_1|1, n__f_1|2])}"
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(4)
BOUNDS(1, n^1)