WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 68 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 0 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:

   f(0) -> cons(0)
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

S is empty.
Rewrite Strategy: FULL
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(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(0) -> 0
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))


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(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:

   f(0) -> cons(0)
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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

   encArg(0) -> 0
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))

Rewrite Strategy: FULL
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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:

   f(0) -> cons(0)
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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

   encArg(0) -> 0
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))

Rewrite Strategy: FULL
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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:

   f(0) -> cons(0)
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
   encArg(0) -> 0
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))

S is empty.
Rewrite Strategy: FULL
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(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 3. 
The certificate found is represented by the following graph.

"[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
{(9,10,[f_1|0, p_1|0, encArg_1|0, encode_f_1|0, encode_0|0, encode_cons_1|0, encode_s_1|0, encode_p_1|0, 0|1, 0|2]), (9,11,[cons_1|1, s_1|1, f_1|1, p_1|1]), (9,12,[f_1|1]), (9,15,[cons_1|2]), (9,16,[f_1|2]), (9,19,[cons_1|3]), (10,10,[0|0, cons_1|0, s_1|0, cons_f_1|0, cons_p_1|0]), (11,10,[0|1, encArg_1|1, 0|2]), (11,11,[cons_1|1, s_1|1, f_1|1, p_1|1]), (11,15,[cons_1|2]), (11,16,[f_1|2]), (11,19,[cons_1|3]), (12,13,[p_1|1]), (12,10,[0|2]), (13,14,[s_1|1]), (14,10,[0|1]), (15,10,[0|2]), (16,17,[p_1|2]), (16,10,[0|3]), (17,18,[s_1|2]), (18,10,[0|2]), (19,10,[0|3])}"
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(8)
BOUNDS(1, n^1)