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


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (4) BOUNDS(1, n^1)
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) SlicingProof [LOWER BOUND(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), 675 ms]
    (14) proven lower bound
    (15) LowerBoundPropagationProof [FINISHED, 0 ms]
    (16) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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

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

(2)
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__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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

(3) 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.

"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
{(1,2,[a__f_1|0, a__c_1|0, a__h_1|0, mark_1|0, f_1|1, d_1|1, c_1|1, h_1|1, a__c_1|1, g_1|1, d_1|2, c_1|2]), (1,3,[a__c_1|1, d_1|2, c_1|2]), (1,6,[a__c_1|1, d_1|2, c_1|2]), (1,7,[a__f_1|1, f_1|2]), (1,8,[a__h_1|1, h_1|2]), (1,9,[a__c_1|2, d_1|3, c_1|3]), (1,10,[a__c_1|2, d_1|3, c_1|3]), (2,2,[f_1|0, g_1|0, d_1|0, c_1|0, h_1|0]), (3,4,[f_1|1]), (4,5,[g_1|1]), (5,2,[f_1|1]), (6,2,[d_1|1]), (7,2,[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]), (7,7,[a__f_1|1, f_1|2]), (7,8,[a__h_1|1, h_1|2]), (7,9,[a__c_1|2, d_1|3, c_1|3]), (7,10,[a__c_1|2, d_1|3, c_1|3]), (8,2,[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]), (8,7,[a__f_1|1, f_1|2]), (8,8,[a__h_1|1, h_1|2]), (8,9,[a__c_1|2, d_1|3, c_1|3]), (8,10,[a__c_1|2, d_1|3, c_1|3]), (9,8,[d_1|2]), (10,11,[f_1|2]), (11,12,[g_1|2]), (12,7,[f_1|2])}"
----------------------------------------

(4)
BOUNDS(1, n^1)

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

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

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


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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

(7) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
a__c/0
g/0
d/0
c/0

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

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


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c
   a__c -> d
   a__h(X) -> a__c
   mark(f(X)) -> a__f(mark(X))
   mark(c) -> a__c
   mark(h(X)) -> a__h(mark(X))
   mark(g) -> g
   mark(d) -> d
   a__f(X) -> f(X)
   a__c -> c
   a__h(X) -> h(X)

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

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

(10)
Obligation:
Innermost TRS:
Rules:
a__f(f(X)) -> a__c
a__c -> d
a__h(X) -> a__c
mark(f(X)) -> a__f(mark(X))
mark(c) -> a__c
mark(h(X)) -> a__h(mark(X))
mark(g) -> g
mark(d) -> d
a__f(X) -> f(X)
a__c -> c
a__h(X) -> h(X)

Types:
a__f :: f:d:c:h:g -> f:d:c:h:g
f :: f:d:c:h:g -> f:d:c:h:g
a__c :: f:d:c:h:g
d :: f:d:c:h:g
a__h :: f:d:c:h:g -> f:d:c:h:g
mark :: f:d:c:h:g -> f:d:c:h:g
c :: f:d:c:h:g
h :: f:d:c:h:g -> f:d:c:h:g
g :: f:d:c:h:g
hole_f:d:c:h:g1_0 :: f:d:c:h:g
gen_f:d:c:h:g2_0 :: Nat -> f:d:c:h:g

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
mark
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
a__f(f(X)) -> a__c
a__c -> d
a__h(X) -> a__c
mark(f(X)) -> a__f(mark(X))
mark(c) -> a__c
mark(h(X)) -> a__h(mark(X))
mark(g) -> g
mark(d) -> d
a__f(X) -> f(X)
a__c -> c
a__h(X) -> h(X)

Types:
a__f :: f:d:c:h:g -> f:d:c:h:g
f :: f:d:c:h:g -> f:d:c:h:g
a__c :: f:d:c:h:g
d :: f:d:c:h:g
a__h :: f:d:c:h:g -> f:d:c:h:g
mark :: f:d:c:h:g -> f:d:c:h:g
c :: f:d:c:h:g
h :: f:d:c:h:g -> f:d:c:h:g
g :: f:d:c:h:g
hole_f:d:c:h:g1_0 :: f:d:c:h:g
gen_f:d:c:h:g2_0 :: Nat -> f:d:c:h:g


Generator Equations:
gen_f:d:c:h:g2_0(0) <=> d
gen_f:d:c:h:g2_0(+(x, 1)) <=> f(gen_f:d:c:h:g2_0(x))


The following defined symbols remain to be analysed:
mark
----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_f:d:c:h:g2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
mark(gen_f:d:c:h:g2_0(+(1, 0)))

Induction Step:
mark(gen_f:d:c:h:g2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
a__f(mark(gen_f:d:c:h:g2_0(+(1, n4_0)))) ->_IH
a__f(*3_0)

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

(14)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a__f(f(X)) -> a__c
a__c -> d
a__h(X) -> a__c
mark(f(X)) -> a__f(mark(X))
mark(c) -> a__c
mark(h(X)) -> a__h(mark(X))
mark(g) -> g
mark(d) -> d
a__f(X) -> f(X)
a__c -> c
a__h(X) -> h(X)

Types:
a__f :: f:d:c:h:g -> f:d:c:h:g
f :: f:d:c:h:g -> f:d:c:h:g
a__c :: f:d:c:h:g
d :: f:d:c:h:g
a__h :: f:d:c:h:g -> f:d:c:h:g
mark :: f:d:c:h:g -> f:d:c:h:g
c :: f:d:c:h:g
h :: f:d:c:h:g -> f:d:c:h:g
g :: f:d:c:h:g
hole_f:d:c:h:g1_0 :: f:d:c:h:g
gen_f:d:c:h:g2_0 :: Nat -> f:d:c:h:g


Generator Equations:
gen_f:d:c:h:g2_0(0) <=> d
gen_f:d:c:h:g2_0(+(x, 1)) <=> f(gen_f:d:c:h:g2_0(x))


The following defined symbols remain to be analysed:
mark
----------------------------------------

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

(16)
BOUNDS(n^1, INF)