WORST_CASE(?,O(n^1))
proof of input_U65lz38jLt.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(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(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(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, 11, 12]
{(1,2,[a__f_1|0, a__c_1|0, a__h_1|0, mark_1|0, f_1|1, d_1|1, c_1|1, h_1|1, a__c_1|1, g_1|1, d_1|2, c_1|2]), (1,3,[a__c_1|1, d_1|2, c_1|2]), (1,6,[a__c_1|1, d_1|2, c_1|2]), (1,7,[a__f_1|1, f_1|2]), (1,8,[a__h_1|1, h_1|2]), (1,9,[a__c_1|2, d_1|3, c_1|3]), (1,10,[a__c_1|2, d_1|3, c_1|3]), (2,2,[f_1|0, g_1|0, d_1|0, c_1|0, h_1|0]), (3,4,[f_1|1]), (4,5,[g_1|1]), (5,2,[f_1|1]), (6,2,[d_1|1]), (7,2,[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]), (7,7,[a__f_1|1, f_1|2]), (7,8,[a__h_1|1, h_1|2]), (7,9,[a__c_1|2, d_1|3, c_1|3]), (7,10,[a__c_1|2, d_1|3, c_1|3]), (8,2,[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]), (8,7,[a__f_1|1, f_1|2]), (8,8,[a__h_1|1, h_1|2]), (8,9,[a__c_1|2, d_1|3, c_1|3]), (8,10,[a__c_1|2, d_1|3, c_1|3]), (9,8,[d_1|2]), (10,11,[f_1|2]), (11,12,[g_1|2]), (12,7,[f_1|2])}"
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(4)
BOUNDS(1, n^1)