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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 201 ms]
    (22) proven lower bound
    (23) LowerBoundPropagationProof [FINISHED, 0 ms]
    (24) BOUNDS(n^1, INF)


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(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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   OR(true, z0) -> c
   OR(z0, true) -> c1
   OR(false, false) -> c2
   MEM(z0, nil) -> c3
   MEM(z0, set(z1)) -> c4
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
S tuples:
   OR(true, z0) -> c
   OR(z0, true) -> c1
   OR(false, false) -> c2
   MEM(z0, nil) -> c3
   MEM(z0, set(z1)) -> c4
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   OR_2, MEM_2

Compound Symbols:   c, c1, c2, c3, c4, c5_3


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   MEM(z0, nil) -> c3
   OR(z0, true) -> c1
   OR(false, false) -> c2
   MEM(z0, set(z1)) -> c4
   OR(true, z0) -> c

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_3


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_2


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_2


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(MEM(x_1, x_2)) = x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(union(x_1, x_2)) = [1] + x_1 + x_2

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
S tuples:none
K tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_2


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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(12)
BOUNDS(1, 1)

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(13) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(14)
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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> ='(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

S is empty.
Rewrite Strategy: INNERMOST
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(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
mem/0
set/0
='/0
='/1

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(16)
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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(nil) -> false
   mem(set) -> ='
   mem(union(y, z)) -> or(mem(y), mem(z))

S is empty.
Rewrite Strategy: INNERMOST
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(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(18)
Obligation:
Innermost TRS:
Rules:
or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(nil) -> false
mem(set) -> ='
mem(union(y, z)) -> or(mem(y), mem(z))

Types:
or :: true:false:=' -> true:false:=' -> true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union -> true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union -> nil:set:union -> nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat -> nil:set:union

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(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
mem
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(20)
Obligation:
Innermost TRS:
Rules:
or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(nil) -> false
mem(set) -> ='
mem(union(y, z)) -> or(mem(y), mem(z))

Types:
or :: true:false:=' -> true:false:=' -> true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union -> true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union -> nil:set:union -> nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat -> nil:set:union


Generator Equations:
gen_nil:set:union3_0(0) <=> nil
gen_nil:set:union3_0(+(x, 1)) <=> union(nil, gen_nil:set:union3_0(x))


The following defined symbols remain to be analysed:
mem
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mem(gen_nil:set:union3_0(n5_0)) -> false, rt in Omega(1 + n5_0)

Induction Base:
mem(gen_nil:set:union3_0(0)) ->_R^Omega(1)
false

Induction Step:
mem(gen_nil:set:union3_0(+(n5_0, 1))) ->_R^Omega(1)
or(mem(nil), mem(gen_nil:set:union3_0(n5_0))) ->_R^Omega(1)
or(false, mem(gen_nil:set:union3_0(n5_0))) ->_IH
or(false, false) ->_R^Omega(1)
false

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

(22)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(nil) -> false
mem(set) -> ='
mem(union(y, z)) -> or(mem(y), mem(z))

Types:
or :: true:false:=' -> true:false:=' -> true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union -> true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union -> nil:set:union -> nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat -> nil:set:union


Generator Equations:
gen_nil:set:union3_0(0) <=> nil
gen_nil:set:union3_0(+(x, 1)) <=> union(nil, gen_nil:set:union3_0(x))


The following defined symbols remain to be analysed:
mem
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(23) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(24)
BOUNDS(n^1, INF)