/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 76 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 4 ms]
(8) BOUNDS(1, n^1)


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

(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(h(x)) -> f(i(x))
   g(i(x)) -> g(h(x))
   h(a) -> b
   i(a) -> b

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(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_b -> b


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

(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(h(x)) -> f(i(x))
   g(i(x)) -> g(h(x))
   h(a) -> b
   i(a) -> b

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_b -> b

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:

   f(h(x)) -> f(i(x))
   g(i(x)) -> g(h(x))
   h(a) -> b
   i(a) -> b

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_b -> b

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:

   f(h(x)) -> f(i(x))
   g(i(x)) -> g(h(x))
   h(a) -> b
   i(a) -> b
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_b -> b

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 2. 
The certificate found is represented by the following graph.

"[62, 63, 64, 65, 66]
{(62,63,[f_1|0, g_1|0, h_1|0, i_1|0, encArg_1|0, encode_f_1|0, encode_h_1|0, encode_i_1|0, encode_g_1|0, encode_a|0, encode_b|0, b|1, a|1, b|2]), (62,64,[f_1|1, g_1|1, h_1|1, i_1|1]), (62,65,[f_1|2]), (62,66,[g_1|2]), (63,63,[a|0, b|0, cons_f_1|0, cons_g_1|0, cons_h_1|0, cons_i_1|0]), (64,63,[encArg_1|1, a|1, b|1, b|2]), (64,64,[f_1|1, g_1|1, h_1|1, i_1|1]), (64,65,[f_1|2]), (64,66,[g_1|2]), (65,64,[i_1|2]), (65,63,[b|2]), (66,64,[h_1|2]), (66,63,[b|2])}"
----------------------------------------

(8)
BOUNDS(1, n^1)