WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 83 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 17 ms]
(8) BOUNDS(1, n^1)


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(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a__g(X) -> a__h(X)
   a__c -> d
   a__h(d) -> a__g(c)
   mark(g(X)) -> a__g(X)
   mark(h(X)) -> a__h(X)
   mark(c) -> a__c
   mark(d) -> d
   a__g(X) -> g(X)
   a__h(X) -> h(X)
   a__c -> c

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(d) -> d
   encArg(c) -> c
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
   encArg(cons_a__c) -> a__c
   encArg(cons_a__h(x_1)) -> a__h(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__g(x_1) -> a__g(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_a__c -> a__c
   encode_d -> d
   encode_c -> c
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a__g(X) -> a__h(X)
   a__c -> d
   a__h(d) -> a__g(c)
   mark(g(X)) -> a__g(X)
   mark(h(X)) -> a__h(X)
   mark(c) -> a__c
   mark(d) -> d
   a__g(X) -> g(X)
   a__h(X) -> h(X)
   a__c -> c

The (relative) TRS S consists of the following rules:

   encArg(d) -> d
   encArg(c) -> c
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
   encArg(cons_a__c) -> a__c
   encArg(cons_a__h(x_1)) -> a__h(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__g(x_1) -> a__g(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_a__c -> a__c
   encode_d -> d
   encode_c -> c
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a__g(X) -> a__h(X)
   a__c -> d
   a__h(d) -> a__g(c)
   mark(g(X)) -> a__g(X)
   mark(h(X)) -> a__h(X)
   mark(c) -> a__c
   mark(d) -> d
   a__g(X) -> g(X)
   a__h(X) -> h(X)
   a__c -> c

The (relative) TRS S consists of the following rules:

   encArg(d) -> d
   encArg(c) -> c
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
   encArg(cons_a__c) -> a__c
   encArg(cons_a__h(x_1)) -> a__h(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__g(x_1) -> a__g(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_a__c -> a__c
   encode_d -> d
   encode_c -> c
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

(6)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a__g(X) -> a__h(X)
   a__c -> d
   a__h(d) -> a__g(c)
   mark(g(X)) -> a__g(X)
   mark(h(X)) -> a__h(X)
   mark(c) -> a__c
   mark(d) -> d
   a__g(X) -> g(X)
   a__h(X) -> h(X)
   a__c -> c
   encArg(d) -> d
   encArg(c) -> c
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
   encArg(cons_a__c) -> a__c
   encArg(cons_a__h(x_1)) -> a__h(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__g(x_1) -> a__g(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_a__c -> a__c
   encode_d -> d
   encode_c -> c
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))

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

(7) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 6. 
The certificate found is represented by the following graph.

"[9, 10, 11, 12, 15, 16, 28]
{(9,10,[a__g_1|0, a__c|0, a__h_1|0, mark_1|0, encArg_1|0, encode_a__g_1|0, encode_a__h_1|0, encode_a__c|0, encode_d|0, encode_c|0, encode_mark_1|0, encode_g_1|0, encode_h_1|0, a__h_1|1, g_1|1, d|1, c|1, h_1|1, a__g_1|1, a__c|1, a__h_1|2, g_1|2, d|2, c|2, h_1|2, h_1|3, a__c|2, d|3, c|3]), (9,11,[a__g_1|1, a__h_1|2, g_1|2, h_1|3]), (9,12,[g_1|1, h_1|1, a__g_1|1, a__h_1|1, mark_1|1, a__h_1|2, g_1|2, h_1|2, h_1|3, a__g_1|2, a__h_1|3, g_1|3, h_1|4]), (9,15,[a__g_1|2, a__h_1|3, g_1|3, h_1|4, a__h_1|2, h_1|3]), (9,16,[a__g_1|3, a__h_1|4, g_1|4, a__g_1|2, h_1|5, a__h_1|2, a__h_1|3, g_1|3, h_1|3, h_1|4]), (9,28,[a__g_1|4, a__h_1|5, g_1|5, a__g_1|2, h_1|6, a__h_1|2, a__h_1|3, g_1|3, h_1|3, h_1|4]), (10,10,[d|0, c|0, g_1|0, h_1|0, cons_a__g_1|0, cons_a__c|0, cons_a__h_1|0, cons_mark_1|0]), (11,10,[c|1]), (12,10,[encArg_1|1, d|1, c|1, a__c|1, d|2, c|2, a__c|2, d|3, c|3]), (12,12,[g_1|1, h_1|1, a__g_1|1, a__h_1|1, mark_1|1, a__h_1|2, g_1|2, h_1|2, a__g_1|2, a__h_1|3, g_1|3, h_1|3, h_1|4]), (12,15,[a__g_1|2, a__h_1|3, g_1|3, h_1|4, a__h_1|2, h_1|3]), (12,16,[a__g_1|3, a__h_1|4, g_1|4, a__g_1|2, h_1|5, a__h_1|2, a__h_1|3, g_1|3, h_1|3, h_1|4]), (12,28,[a__g_1|4, a__h_1|5, g_1|5, a__g_1|2, h_1|6, a__h_1|2, a__h_1|3, g_1|3, h_1|3, h_1|4]), (15,10,[c|2]), (16,10,[c|3]), (28,10,[c|4])}"
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(8)
BOUNDS(1, n^1)