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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 59 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 10 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:

   p(f(f(x))) -> q(f(g(x)))
   p(g(g(x))) -> q(g(f(x)))
   q(f(f(x))) -> p(f(g(x)))
   q(g(g(x))) -> p(g(f(x)))

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(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))
   encode_g(x_1) -> g(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:

   p(f(f(x))) -> q(f(g(x)))
   p(g(g(x))) -> q(g(f(x)))
   q(f(f(x))) -> p(f(g(x)))
   q(g(g(x))) -> p(g(f(x)))

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))
   encode_g(x_1) -> g(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:

   p(f(f(x))) -> q(f(g(x)))
   p(g(g(x))) -> q(g(f(x)))
   q(f(f(x))) -> p(f(g(x)))
   q(g(g(x))) -> p(g(f(x)))

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))
   encode_g(x_1) -> g(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:

   p(f(f(x))) -> q(f(g(x)))
   p(g(g(x))) -> q(g(f(x)))
   q(f(f(x))) -> p(f(g(x)))
   q(g(g(x))) -> p(g(f(x)))
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))
   encode_g(x_1) -> g(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 2. 
The certificate found is represented by the following graph.

"[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]
{(5,6,[p_1|0, q_1|0, encArg_1|0, encode_p_1|0, encode_f_1|0, encode_q_1|0, encode_g_1|0]), (5,7,[q_1|1]), (5,9,[q_1|1]), (5,11,[p_1|1]), (5,13,[p_1|1]), (5,15,[f_1|1, g_1|1, p_1|1, q_1|1]), (5,16,[q_1|2]), (5,18,[q_1|2]), (5,20,[p_1|2]), (5,22,[p_1|2]), (6,6,[f_1|0, g_1|0, cons_p_1|0, cons_q_1|0]), (7,8,[f_1|1]), (8,6,[g_1|1]), (9,10,[g_1|1]), (10,6,[f_1|1]), (11,12,[f_1|1]), (12,6,[g_1|1]), (13,14,[g_1|1]), (14,6,[f_1|1]), (15,6,[encArg_1|1]), (15,15,[f_1|1, g_1|1, p_1|1, q_1|1]), (15,16,[q_1|2]), (15,18,[q_1|2]), (15,20,[p_1|2]), (15,22,[p_1|2]), (16,17,[f_1|2]), (17,15,[g_1|2]), (18,19,[g_1|2]), (19,15,[f_1|2]), (20,21,[f_1|2]), (21,15,[g_1|2]), (22,23,[g_1|2]), (23,15,[f_1|2])}"
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(8)
BOUNDS(1, n^1)