/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 5 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 412 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 98 ms]
            (20) typed CpxTrs


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

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


The TRS R consists of the following rules:

   active(f(X)) -> mark(g(h(f(X))))
   active(f(X)) -> f(active(X))
   active(h(X)) -> h(active(X))
   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   proper(f(X)) -> f(proper(X))
   proper(g(X)) -> g(proper(X))
   proper(h(X)) -> h(proper(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))

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

(2)
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(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

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

(4)
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(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(5) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[25, 26, 27, 28, 29, 30, 31, 32, 33]
{(25,26,[f_1|0, h_1|0, g_1|0, top_1|0]), (25,27,[mark_1|1]), (25,28,[ok_1|1]), (25,29,[mark_1|1]), (25,30,[ok_1|1]), (25,31,[ok_1|1]), (25,32,[top_1|1]), (25,33,[top_1|1]), (26,26,[mark_1|0, ok_1|0, proper_1|0, active_1|0]), (27,26,[f_1|1]), (27,27,[mark_1|1]), (27,28,[ok_1|1]), (28,26,[f_1|1]), (28,27,[mark_1|1]), (28,28,[ok_1|1]), (29,26,[h_1|1]), (29,29,[mark_1|1]), (29,30,[ok_1|1]), (30,26,[h_1|1]), (30,29,[mark_1|1]), (30,30,[ok_1|1]), (31,26,[g_1|1]), (31,31,[ok_1|1]), (32,26,[proper_1|1]), (33,26,[active_1|1])}"
----------------------------------------

(6)
BOUNDS(1, n^1)

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

(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(f(X)) -> mark(g(h(f(X))))
   active(f(X)) -> f(active(X))
   active(h(X)) -> h(active(X))
   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   proper(f(X)) -> f(proper(X))
   proper(g(X)) -> g(proper(X))
   proper(h(X)) -> h(proper(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
TRS:
Rules:
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
h(mark(X)) -> mark(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
g :: mark:ok -> mark:ok
h :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, g, h, f, proper, top

They will be analysed ascendingly in the following order:
g < active
h < active
f < active
active < top
g < proper
h < proper
f < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
h(mark(X)) -> mark(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
g :: mark:ok -> mark:ok
h :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok


Generator Equations:
gen_mark:ok3_0(0) <=> hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) <=> mark(gen_mark:ok3_0(x))


The following defined symbols remain to be analysed:
g, active, h, f, proper, top

They will be analysed ascendingly in the following order:
g < active
h < active
f < active
active < top
g < proper
h < proper
f < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_mark:ok3_0(+(1, n9_0))) -> *4_0, rt in Omega(n9_0)

Induction Base:
h(gen_mark:ok3_0(+(1, 0)))

Induction Step:
h(gen_mark:ok3_0(+(1, +(n9_0, 1)))) ->_R^Omega(1)
mark(h(gen_mark:ok3_0(+(1, n9_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
h(mark(X)) -> mark(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
g :: mark:ok -> mark:ok
h :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok


Generator Equations:
gen_mark:ok3_0(0) <=> hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) <=> mark(gen_mark:ok3_0(x))


The following defined symbols remain to be analysed:
h, active, f, proper, top

They will be analysed ascendingly in the following order:
h < active
f < active
active < top
h < proper
f < proper
proper < top

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

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
h(mark(X)) -> mark(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
g :: mark:ok -> mark:ok
h :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok


Lemmas:
h(gen_mark:ok3_0(+(1, n9_0))) -> *4_0, rt in Omega(n9_0)


Generator Equations:
gen_mark:ok3_0(0) <=> hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) <=> mark(gen_mark:ok3_0(x))


The following defined symbols remain to be analysed:
f, active, proper, top

They will be analysed ascendingly in the following order:
f < active
active < top
f < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_mark:ok3_0(+(1, n315_0))) -> *4_0, rt in Omega(n315_0)

Induction Base:
f(gen_mark:ok3_0(+(1, 0)))

Induction Step:
f(gen_mark:ok3_0(+(1, +(n315_0, 1)))) ->_R^Omega(1)
mark(f(gen_mark:ok3_0(+(1, n315_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
h(mark(X)) -> mark(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
g :: mark:ok -> mark:ok
h :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok


Lemmas:
h(gen_mark:ok3_0(+(1, n9_0))) -> *4_0, rt in Omega(n9_0)
f(gen_mark:ok3_0(+(1, n315_0))) -> *4_0, rt in Omega(n315_0)


Generator Equations:
gen_mark:ok3_0(0) <=> hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) <=> mark(gen_mark:ok3_0(x))


The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top