WORST_CASE(?,O(n^1))
proof of input_3gKBMSn6i7.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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)


----------------------------------------

(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(X) -> g(n__h(n__f(X)))
   h(X) -> n__h(X)
   f(X) -> n__f(X)
   activate(n__h(X)) -> h(activate(X))
   activate(n__f(X)) -> f(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(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(X) -> g(n__h(n__f(X)))
   h(X) -> n__h(X)
   f(X) -> n__f(X)
   activate(n__h(X)) -> h(activate(X))
   activate(n__f(X)) -> f(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(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, 7, 8]
{(1,2,[f_1|0, h_1|0, activate_1|0, n__f_1|1, n__h_1|1, g_1|1]), (1,3,[g_1|1]), (1,5,[h_1|1, n__h_1|2]), (1,6,[f_1|1, n__f_1|2]), (1,7,[g_1|2]), (2,2,[g_1|0, n__h_1|0, n__f_1|0]), (3,4,[n__h_1|1]), (4,2,[n__f_1|1]), (5,2,[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]), (5,5,[h_1|1, n__h_1|2]), (5,6,[f_1|1, n__f_1|2]), (5,7,[g_1|2]), (6,2,[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]), (6,5,[h_1|1, n__h_1|2]), (6,6,[f_1|1, n__f_1|2]), (6,7,[g_1|2]), (7,8,[n__h_1|2]), (8,6,[n__f_1|2])}"
----------------------------------------

(4)
BOUNDS(1, n^1)