WORST_CASE(?,O(n^1))
proof of input_S4QbQIAop6.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(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(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(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.

"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
{(1,2,[f_1|0, e_1|0, a|1, c_1|1, d_1|1, b|1, g_1|1, e_1|1, a|2, b|2, b|3]), (1,3,[f_1|1]), (1,5,[f_1|1]), (1,7,[f_1|1]), (1,9,[f_1|2]), (2,2,[a|0, c_1|0, d_1|0, b|0, g_1|0]), (3,4,[c_1|1]), (4,2,[a|1]), (5,6,[d_1|1]), (6,2,[a|1]), (7,8,[d_1|1]), (8,2,[b|1]), (9,10,[d_1|2]), (10,2,[b|2])}"
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(4)
BOUNDS(1, n^1)