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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 213 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), 196 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^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:

   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, y, y) -> y
   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
   if(x, if(x, y, z), z) -> if(x, y, z)
   if(x, y, if(x, y, z)) -> if(x, y, z)

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(true) -> true
   encArg(false) -> false
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false


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

(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:

   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, y, y) -> y
   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
   if(x, if(x, y, z), z) -> if(x, y, z)
   if(x, y, if(x, y, z)) -> if(x, y, z)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false

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:

   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, y, y) -> y
   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
   if(x, if(x, y, z), z) -> if(x, y, z)
   if(x, y, if(x, y, z)) -> if(x, y, z)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false

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:

   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, y, y) -> y
   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
   if(x, if(x, y, z), z) -> if(x, y, z)
   if(x, y, if(x, y, z)) -> if(x, y, z)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false

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

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

(8)
Obligation:
Innermost TRS:
Rules:
if(true, x, y) -> x
if(false, x, y) -> y
if(x, y, y) -> y
if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
if(x, if(x, y, z), z) -> if(x, y, z)
if(x, y, if(x, y, z)) -> if(x, y, z)
encArg(true) -> true
encArg(false) -> false
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_true -> true
encode_false -> false

Types:
if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
true :: true:false:cons_if
false :: true:false:cons_if
encArg :: true:false:cons_if -> true:false:cons_if
cons_if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
encode_if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
encode_true :: true:false:cons_if
encode_false :: true:false:cons_if
hole_true:false:cons_if1_0 :: true:false:cons_if
gen_true:false:cons_if2_0 :: Nat -> true:false:cons_if

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

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

They will be analysed ascendingly in the following order:
if < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
if(true, x, y) -> x
if(false, x, y) -> y
if(x, y, y) -> y
if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
if(x, if(x, y, z), z) -> if(x, y, z)
if(x, y, if(x, y, z)) -> if(x, y, z)
encArg(true) -> true
encArg(false) -> false
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_true -> true
encode_false -> false

Types:
if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
true :: true:false:cons_if
false :: true:false:cons_if
encArg :: true:false:cons_if -> true:false:cons_if
cons_if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
encode_if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
encode_true :: true:false:cons_if
encode_false :: true:false:cons_if
hole_true:false:cons_if1_0 :: true:false:cons_if
gen_true:false:cons_if2_0 :: Nat -> true:false:cons_if


Generator Equations:
gen_true:false:cons_if2_0(0) <=> true
gen_true:false:cons_if2_0(+(x, 1)) <=> cons_if(true, true, gen_true:false:cons_if2_0(x))


The following defined symbols remain to be analysed:
if, encArg

They will be analysed ascendingly in the following order:
if < encArg

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_true:false:cons_if2_0(n106_0)) -> gen_true:false:cons_if2_0(0), rt in Omega(n106_0)

Induction Base:
encArg(gen_true:false:cons_if2_0(0)) ->_R^Omega(0)
true

Induction Step:
encArg(gen_true:false:cons_if2_0(+(n106_0, 1))) ->_R^Omega(0)
if(encArg(true), encArg(true), encArg(gen_true:false:cons_if2_0(n106_0))) ->_R^Omega(0)
if(true, encArg(true), encArg(gen_true:false:cons_if2_0(n106_0))) ->_R^Omega(0)
if(true, true, encArg(gen_true:false:cons_if2_0(n106_0))) ->_IH
if(true, true, gen_true:false:cons_if2_0(0)) ->_R^Omega(1)
true

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

(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
if(true, x, y) -> x
if(false, x, y) -> y
if(x, y, y) -> y
if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
if(x, if(x, y, z), z) -> if(x, y, z)
if(x, y, if(x, y, z)) -> if(x, y, z)
encArg(true) -> true
encArg(false) -> false
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_true -> true
encode_false -> false

Types:
if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
true :: true:false:cons_if
false :: true:false:cons_if
encArg :: true:false:cons_if -> true:false:cons_if
cons_if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
encode_if :: true:false:cons_if -> true:false:cons_if -> true:false:cons_if -> true:false:cons_if
encode_true :: true:false:cons_if
encode_false :: true:false:cons_if
hole_true:false:cons_if1_0 :: true:false:cons_if
gen_true:false:cons_if2_0 :: Nat -> true:false:cons_if


Generator Equations:
gen_true:false:cons_if2_0(0) <=> true
gen_true:false:cons_if2_0(+(x, 1)) <=> cons_if(true, true, gen_true:false:cons_if2_0(x))


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

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

(14)
BOUNDS(n^1, INF)