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


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 201 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 991 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 727 ms]
        (18) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

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

(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(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

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

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

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

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

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

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
----------------------------------------

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

(8)
Obligation:
Innermost TRS:
Rules:
a__f(X) -> a__if(mark(X), c, f(true))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(f(X)) -> a__f(mark(X))
mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
a__f(X) -> f(X)
a__if(X1, X2, X3) -> if(X1, X2, X3)
encArg(c) -> c
encArg(f(x_1)) -> f(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1) -> a__f(encArg(x_1))
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_mark(x_1) -> mark(encArg(x_1))
encode_c -> c
encode_f(x_1) -> f(encArg(x_1))
encode_true -> true
encode_false -> false
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
c :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
true :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
false :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encArg :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_c :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_true :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_false :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
hole_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark1_0 :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0 :: Nat -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__f, a__if, mark, encArg

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__f < encArg
a__if = mark
a__if < encArg
mark < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
a__f(X) -> a__if(mark(X), c, f(true))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(f(X)) -> a__f(mark(X))
mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
a__f(X) -> f(X)
a__if(X1, X2, X3) -> if(X1, X2, X3)
encArg(c) -> c
encArg(f(x_1)) -> f(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1) -> a__f(encArg(x_1))
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_mark(x_1) -> mark(encArg(x_1))
encode_c -> c
encode_f(x_1) -> f(encArg(x_1))
encode_true -> true
encode_false -> false
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
c :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
true :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
false :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encArg :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_c :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_true :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_false :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
hole_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark1_0 :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0 :: Nat -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark


Generator Equations:
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(0) <=> c
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(x, 1)) <=> f(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(x))


The following defined symbols remain to be analysed:
a__if, a__f, mark, encArg

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__f < encArg
a__if = mark
a__if < encArg
mark < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(1, n26_0))) -> *3_0, rt in Omega(n26_0)

Induction Base:
mark(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(1, 0)))

Induction Step:
mark(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(1, +(n26_0, 1)))) ->_R^Omega(1)
a__f(mark(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(1, n26_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).
----------------------------------------

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a__f(X) -> a__if(mark(X), c, f(true))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(f(X)) -> a__f(mark(X))
mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
a__f(X) -> f(X)
a__if(X1, X2, X3) -> if(X1, X2, X3)
encArg(c) -> c
encArg(f(x_1)) -> f(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1) -> a__f(encArg(x_1))
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_mark(x_1) -> mark(encArg(x_1))
encode_c -> c
encode_f(x_1) -> f(encArg(x_1))
encode_true -> true
encode_false -> false
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
c :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
true :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
false :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encArg :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_c :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_true :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_false :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
hole_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark1_0 :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0 :: Nat -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark


Generator Equations:
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(0) <=> c
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(x, 1)) <=> f(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(x))


The following defined symbols remain to be analysed:
mark, a__f, encArg

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__f < encArg
a__if = mark
a__if < encArg
mark < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
a__f(X) -> a__if(mark(X), c, f(true))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(f(X)) -> a__f(mark(X))
mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
a__f(X) -> f(X)
a__if(X1, X2, X3) -> if(X1, X2, X3)
encArg(c) -> c
encArg(f(x_1)) -> f(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1) -> a__f(encArg(x_1))
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_mark(x_1) -> mark(encArg(x_1))
encode_c -> c
encode_f(x_1) -> f(encArg(x_1))
encode_true -> true
encode_false -> false
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
c :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
true :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
false :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encArg :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
cons_mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_a__f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_a__if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_mark :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_c :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_f :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_true :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_false :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
encode_if :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
hole_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark1_0 :: c:true:f:false:if:cons_a__f:cons_a__if:cons_mark
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0 :: Nat -> c:true:f:false:if:cons_a__f:cons_a__if:cons_mark


Lemmas:
mark(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(1, n26_0))) -> *3_0, rt in Omega(n26_0)


Generator Equations:
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(0) <=> c
gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(x, 1)) <=> f(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(x))


The following defined symbols remain to be analysed:
a__f, a__if, encArg

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__f < encArg
a__if = mark
a__if < encArg
mark < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(n23977_0)) -> gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(n23977_0), rt in Omega(0)

Induction Base:
encArg(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(0)) ->_R^Omega(0)
c

Induction Step:
encArg(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(+(n23977_0, 1))) ->_R^Omega(0)
f(encArg(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(n23977_0))) ->_IH
f(gen_c:true:f:false:if:cons_a__f:cons_a__if:cons_mark2_0(c23978_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(18)
BOUNDS(1, INF)