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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) 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), 45 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 2 ms]
(8) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   a__c -> a__f(g(c))
   a__f(g(X)) -> g(X)
   mark(c) -> a__c
   mark(f(X)) -> a__f(X)
   mark(g(X)) -> g(X)
   a__c -> c
   a__f(X) -> f(X)

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

(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(g(x_1)) -> g(encArg(x_1))
   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(cons_a__c) -> a__c
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_c -> c
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))


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

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


The TRS R consists of the following rules:

   a__c -> a__f(g(c))
   a__f(g(X)) -> g(X)
   mark(c) -> a__c
   mark(f(X)) -> a__f(X)
   mark(g(X)) -> g(X)
   a__c -> c
   a__f(X) -> f(X)

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(cons_a__c) -> a__c
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_c -> c
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   a__c -> a__f(g(c))
   a__f(g(X)) -> g(X)
   mark(c) -> a__c
   mark(f(X)) -> a__f(X)
   mark(g(X)) -> g(X)
   a__c -> c
   a__f(X) -> f(X)

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(cons_a__c) -> a__c
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_c -> c
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   a__c -> a__f(g(c))
   a__f(g(X)) -> g(X)
   mark(c) -> a__c
   mark(f(X)) -> a__f(X)
   mark(g(X)) -> g(X)
   a__c -> c
   a__f(X) -> f(X)
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(cons_a__c) -> a__c
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_c -> c
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))

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

(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 4. 
The certificate found is represented by the following graph.

"[15, 16, 17, 18, 19, 20, 21, 22, 23]
{(15,16,[a__c|0, a__f_1|0, mark_1|0, encArg_1|0, encode_a__c|0, encode_a__f_1|0, encode_g_1|0, encode_c|0, encode_mark_1|0, encode_f_1|0, c|1, f_1|1, g_1|1, a__c|1, a__f_1|1, c|2, f_1|2, a__c|2, c|3]), (15,17,[a__f_1|1, f_1|2]), (15,19,[g_1|1, f_1|1, a__f_1|1, mark_1|1, f_1|2, g_1|2, a__f_1|2, f_1|3, g_1|3]), (15,20,[a__f_1|2, f_1|3]), (15,18,[g_1|2]), (15,21,[g_1|3, g_1|2]), (15,22,[a__f_1|3, f_1|4, a__f_1|2, f_1|3]), (15,23,[g_1|4, g_1|2, g_1|3]), (16,16,[g_1|0, c|0, f_1|0, cons_a__c|0, cons_a__f_1|0, cons_mark_1|0]), (17,18,[g_1|1]), (18,16,[c|1]), (19,16,[encArg_1|1, c|1, a__c|1, c|2, a__c|2, c|3]), (19,19,[g_1|1, f_1|1, a__f_1|1, mark_1|1, f_1|2, g_1|2, a__f_1|2, f_1|3, g_1|3]), (19,20,[a__f_1|2, f_1|3]), (19,22,[a__f_1|3, f_1|4, a__f_1|2, f_1|3]), (19,21,[g_1|3, g_1|2]), (19,23,[g_1|4, g_1|2, g_1|3]), (20,21,[g_1|2]), (21,16,[c|2]), (22,23,[g_1|3]), (23,16,[c|3])}"
----------------------------------------

(8)
BOUNDS(1, n^1)