/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(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 183 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), 258 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:

   and(x, or(y, z)) -> or(and(x, y), and(x, z))
   and(x, and(y, y)) -> and(x, y)
   or(or(x, y), and(y, z)) -> or(x, y)
   or(x, and(x, y)) -> x
   or(true, y) -> true
   or(x, false) -> x
   or(x, x) -> x
   or(x, or(y, y)) -> or(x, y)
   and(x, true) -> x
   and(false, y) -> false
   and(x, x) -> x

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_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   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:

   and(x, or(y, z)) -> or(and(x, y), and(x, z))
   and(x, and(y, y)) -> and(x, y)
   or(or(x, y), and(y, z)) -> or(x, y)
   or(x, and(x, y)) -> x
   or(true, y) -> true
   or(x, false) -> x
   or(x, x) -> x
   or(x, or(y, y)) -> or(x, y)
   and(x, true) -> x
   and(false, y) -> false
   and(x, x) -> x

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   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:

   and(x, or(y, z)) -> or(and(x, y), and(x, z))
   and(x, and(y, y)) -> and(x, y)
   or(or(x, y), and(y, z)) -> or(x, y)
   or(x, and(x, y)) -> x
   or(true, y) -> true
   or(x, false) -> x
   or(x, x) -> x
   or(x, or(y, y)) -> or(x, y)
   and(x, true) -> x
   and(false, y) -> false
   and(x, x) -> x

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   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:

   and(x, or(y, z)) -> or(and(x, y), and(x, z))
   and(x, and(y, y)) -> and(x, y)
   or(or(x, y), and(y, z)) -> or(x, y)
   or(x, and(x, y)) -> x
   or(true, y) -> true
   or(x, false) -> x
   or(x, x) -> x
   or(x, or(y, y)) -> or(x, y)
   and(x, true) -> x
   and(false, y) -> false
   and(x, x) -> x

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

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

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

(8)
Obligation:
Innermost TRS:
Rules:
and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
encArg(true) -> true
encArg(false) -> false
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false

Types:
and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
true :: true:false:cons_and:cons_or
false :: true:false:cons_and:cons_or
encArg :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or
cons_and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
cons_or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_true :: true:false:cons_and:cons_or
encode_false :: true:false:cons_and:cons_or
hole_true:false:cons_and:cons_or1_0 :: true:false:cons_and:cons_or
gen_true:false:cons_and:cons_or2_0 :: Nat -> true:false:cons_and:cons_or

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
encArg(true) -> true
encArg(false) -> false
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false

Types:
and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
true :: true:false:cons_and:cons_or
false :: true:false:cons_and:cons_or
encArg :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or
cons_and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
cons_or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_true :: true:false:cons_and:cons_or
encode_false :: true:false:cons_and:cons_or
hole_true:false:cons_and:cons_or1_0 :: true:false:cons_and:cons_or
gen_true:false:cons_and:cons_or2_0 :: Nat -> true:false:cons_and:cons_or


Generator Equations:
gen_true:false:cons_and:cons_or2_0(0) <=> true
gen_true:false:cons_and:cons_or2_0(+(x, 1)) <=> cons_and(true, gen_true:false:cons_and:cons_or2_0(x))


The following defined symbols remain to be analysed:
or, and, encArg

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_true:false:cons_and:cons_or2_0(n127_0)) -> gen_true:false:cons_and:cons_or2_0(0), rt in Omega(n127_0)

Induction Base:
encArg(gen_true:false:cons_and:cons_or2_0(0)) ->_R^Omega(0)
true

Induction Step:
encArg(gen_true:false:cons_and:cons_or2_0(+(n127_0, 1))) ->_R^Omega(0)
and(encArg(true), encArg(gen_true:false:cons_and:cons_or2_0(n127_0))) ->_R^Omega(0)
and(true, encArg(gen_true:false:cons_and:cons_or2_0(n127_0))) ->_IH
and(true, gen_true:false:cons_and:cons_or2_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:
and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
encArg(true) -> true
encArg(false) -> false
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false

Types:
and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
true :: true:false:cons_and:cons_or
false :: true:false:cons_and:cons_or
encArg :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or
cons_and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
cons_or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_and :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_or :: true:false:cons_and:cons_or -> true:false:cons_and:cons_or -> true:false:cons_and:cons_or
encode_true :: true:false:cons_and:cons_or
encode_false :: true:false:cons_and:cons_or
hole_true:false:cons_and:cons_or1_0 :: true:false:cons_and:cons_or
gen_true:false:cons_and:cons_or2_0 :: Nat -> true:false:cons_and:cons_or


Generator Equations:
gen_true:false:cons_and:cons_or2_0(0) <=> true
gen_true:false:cons_and:cons_or2_0(+(x, 1)) <=> cons_and(true, gen_true:false:cons_and:cons_or2_0(x))


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

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

(14)
BOUNDS(n^1, INF)